Pagina principale Physics and Dance

Physics and Dance

A fascinating exploration of our reality through the eyes of a physicist and a dancer—and an engaging introduction to both disciplines

From stepping out of our beds each morning to admiring the stars at night, we live in a world of motion, energy, space, and time. How do we understand the phenomena that shape our experience? How do we make sense of our physical realities? Two guides—a former member of New York City Ballet, Emily Coates, and a CERN particle physicist, Sarah Demers—show us how their respective disciplines can help us to understand both the quotidian and the deepest questions about the universe. Requiring no previous knowledge of dance or physics, this introduction covers the fundamentals while revealing how a dialogue between art and science can enrich our appreciation of both. Readers will come away with a broad cultural knowledge of Newtonian to quantum mechanics and classical to contemporary dance. Including problem sets and choreographic exercises to solidify understanding, this book will be of interest to anyone curious about physics or dance.
Categories: Physics
Anno: 2019
Editore: Yale University Press
Lingua: english
Pagine: 193
ISBN 10: 0300195834
ISBN 13: 9780300195835
File: PDF, 5.30 MB
Physics and Dance

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Physics and Dance
Emily Coates and Sarah Demers

New Haven and London

Published with assistance from the Alfred P. Sloan Foundation.
Copyright c 2019 by Emily Coates and Sarah Demers.
All rights reserved.
press), without written permission from the publishers.
Yale University Press books may be purchased in quantity for educational,
business, or promotional use. For information, please e-mail (U.S. office) or (U.K. office).
Printed in the United States of America.
Library of Congress Control Number: 2018933407
ISBN 978-0-300-19583-5 (hardcover : alk. paper)
This paper meets the requirements of ANSI/NISO Z39.48-1992
(Permanence of Paper).
10 9 8 7 6 5 4 3 2 1



Part I: Principles of Movement

Tuning in to Gravity
The Universal Law of Gravitation
Center of Mass
Balancing Together
Area of Support
The Dance of the Future
Interdisciplinary Thinking



Newton’s 1st Law of Motion
Newton’s 2nd Law of Motion
The Plié
Down and Up
Free Body Diagrams
A Cultural Force Diagram
Force Plates and the Plié
Newton’s 3rd Law of Motion
How Do We Move?
Everyday Action
When Is Walking More Than Walking?



Dancing Feet
Taking Off and Landing
Projectile Motion
Equations of Constant Acceleration
Jumping Up and Out
Calculating a Jump
Zero Velocity




Improvising with Friction
Kinetic Friction
Static Friction
Calculating Frictional Forces
If People Were Boxes
Composing with Friction
Tango in the World



Catching a Wave
Calculating Momentum
Conservation of Momentum
A Momentum Movement Phrase
Conservation of Momentum in a System
The Momentum of Two (or Three)



The Pirouette
Rotational Motion
The Lunge
The Arms
Moment of Inertia
The Turn
Angular Momentum
Kinesthetic Intelligence


Part II: Energy, Space, Time

Kinetic Energy
Energy as Form
Gravitational Potential Energy
Spring Potential Energy
Mixing Energies
The Energy of Light and the Electromagnetic Spectrum
From Marking to Attack
Quantized Energy and Bohr’s Atom
Energy Images
Dark Energy
Real Versus Apparent Energy
E = mc2




Dancing Cells
A Glimpse into the Microscopic World
The Geography of the Body
Dancers in the Universe
Room Writing
Postulates of Special Relativity
Special Relativity: Length Contraction
Absolute to Relative Choreographic Space
Stage Space



Special Relativity: Time Dilation
Cunningham’s ”Space, Time and Dance”
General Relativity
Extended Duration
The Lifetime of Muons
Rates of Change
Gravitational Waves
The Story of the Universe






Physics Problems
Choreographic Studies







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In certain respects, teaching physics and dance together seems obvious. Both
physicists and dancers spend time imagining, enacting, modeling, and assessing motion. Physicists must identify and quantify the forces that act
upon moving bodies. Dancers must move within a world of physical rules.
Knowing more about these natural forces may help a dancer to understand
dance technique more completely, and moving may help a physicist’s imagination take flight.
Yet as much as they share a basic focus, physics and dance also differ in
significant ways. The disciplines have very different approaches to methods
and outcomes of research, modes of evaluation, and truth claims. Physicists and dancers acquire expertise through a wide range of practices—in
varying degrees mathematical, experiential, embodied, and theoretical, depending on the line of inquiry in any given choreographic project or physics
experiment. Their spatial scales do not always line up: in subatomic physics,
the laws of motion change completely from those of the world available to
human perception, in which choreographers’ imaginations must live. Grappling with the differences between physics and dance can lead practitioners
to new connections and unexpected revelations. The dance studio becomes
a laboratory, and the organizing principles of planetary motion become a
choreographic score.
Readers of this book will encounter a variety of interactions between
physics and dance. In the first part of the book, we pair introductory topics in classical physics with basic principles of dance technique and relevant
dance history. We organize these chapters according to terms common to
both disciplines: gravity, force, motion, friction, momentum, and turning. In
this part of the book, the connections between the disciplines are fairly direct. Physicists separate types of friction into static and kinetic, for instance,
and deploy different coefficients and formulas to quantify the forces accordingly. Dancers recognize this same friction tactilely, as a force that impedes
or enhances movement, giving a dance texture and meaning. The physics
and dance pairings guide the reader through a physical encounter with the
natural forces that physics describes while introducing some of the practices
by which dance artists relate to those forces. We are interested in the multifaceted understanding that can develop from conjoining different ways of
knowing the same thing.


In the second part of the book, we open up the inquiry to energy, space,
and time—three broad concepts upon which the disciplines of physics and
dance rest. In these chapters, we move more deeply into modern physics
and choreographic research. Both have radically altered our perception of
energy, space, and time, but they have done so using significantly different
methods and under significantly different conditions. The realities predicted
by modern physics are imperceptible to humans: we cannot travel at the
speed of light or have a sensorial encounter with a particle. But choreographers can alter an audience’s experience of time and space—in effect enacting relativistic conditions in live performance. Because the terms in which
the research occurs diverge so dramatically, the connections between physics
and dance in these chapters are less direct and not always obvious. As we
weave through the various topics, we create certain links, through analogy
or by tracing common threads. But we also intentionally give readers room
to forge their own correlations and insights.
In addition to introducing readers to physics and dance, this book offers a primer in interdisciplinary research. Over the past seven years, we
have collaborated in many different formats—from co-teaching to writing
to creating art in different media. In that process, we have come to recognize that the nature of the connections we make between our disciplines can
vary, from explicit to implicit, from the smoothly parallel to the seemingly
far-fetched. Analogies might break down quickly or hold up surprisingly
well. The point of interdisciplinary research is to assemble different ways
of understanding the world, in order to ask questions whose answers might
not be accessible through the individual disciplines alone.
Throughout the book we place equal emphasis on problem solving, movement exercises, and choreographic studies, without making hierarchical distinctions. These are practices that deepen and strengthen over time. They
are also all forms of thinking—and the boundaries between aesthetic, mathematical, and embodied reasoning are blurrier than many would assume.
Dance practice can be a form of quantitative research, and calculations can be
a version of choreographic reasoning on the page. The consistent throughline in this book is movement.
Our approach departs from the usual teaching methods in both physics
and dance. Physics textbooks typically present an assortment of hypothetical objects to explain concepts, including frictionless ramps, massless pulleys, ideal springs, and particles in boxes, as well as balls, cannons, athletes, coffee makers, and the occasional ballerina. Simplified shapes—such
as a perfect sphere or cube—can give students a clearer introduction to new
ideas. Concepts in modern physics, which are even further removed from
lived experience, can require thought experiments: for Einstein’s theory of
special relativity, the teacher might use the light of a flashlight reflecting off
a mirror on the ceiling of a moving train.
Imagining movement through idealized material objects allows for simpler calculations. But the prevalence of objects also places the reader at
a remove—not as one who experiences the forces but as one who merely

observes them. Frequently missing from standard physics textbooks is the
recognition that the same forces that act on a box act upon the human body.
A heightened awareness of the physics along with a myriad of political
and cultural forces that shape human experience is precisely the research
of dance.
Dance contests the idea that knowledge can be fixed or transmittable in
books. Instead, the art form relies on a process of embodied transmission.
Dance moves body to body, between individual people and among communities. The process is inherently active: to learn to dance, you need to move.
In the passage from one body to the next, the embodied knowledge of dance,
cognitively wired into muscle memories, slowly spreads.
Knowledge is codified in dance forms, which are styles or genres that
preserve movement vocabularies and philosophies of motion. These details continually evolve as dance forms migrate across geographic distances
and historical time periods. Take the many strains of Kathak, a form of Indian classical dance characterized by physical storytelling that originated
centuries ago in northern India and is still performed today. Or in a very
different example, George Balanchine’s neoclassical ballet, in which he synthesized the cultural influences of nineteenth-century Imperial Russia and
mid-twentieth-century America, specifically in New York City. The physical
laws of our planet do not change. But the dancer’s physical, psychological,
and emotional relationship to those laws does change, depending on who
is dancing, who is watching, what is being danced, and where the dance
Dance shows up in the archives through different forms of writing by
artists and scholars. Systems of choreographic notation have cropped up
over the centuries, often grounded in the biomechanical knowledge of the
day. These systems always fall short of effectively preserving all the details of a dance in print. Dance has been documented on film and video—
and these have also become a means of its transmission. Still, performance
and dance scholars grapple with the ephemeral nature of the art form. Unlike standardized models in physics that hold up over many experiments, a
dance will never occur the same way twice.
With such striking differences in the nature of physics and dance knowledge, what are the benefits of studying these disciplines together? The answer depends on your perspective. For science teachers, replacing boxes
and pulleys with the physical experimentation honed by dance gives students different access to the forces studied—allowing them to move from
what they know (moving through space) to what they may not know (the
scientific analysis of movement). Analyzing the motion of dancing bodies
requires a more sophisticated understanding of physics than analyzing the
motion of boxes. For dancers, a more nuanced understanding of physical
conditions can inform dance technique, infusing the dancer’s thought patterns with a new awareness. Images drawn from physics can also inspire
the kinesthetic imagination. For instance, imagining contact with the floor
as an interaction between springs—which is the metaphor we use to explain

Newton’s 3rd Law of Motion—will change the way the movement appears
to the viewer and feels to the dancer.
Other creative and scientific benefits can also accrue. Using physics, a
choreographer may gain new points of departure from which to launch into
motion, and an expanded toolkit of prompts with which to experiment with
energy, space, and time in choreographic composition. Meanwhile, the scientist gains a more multifaceted outlook on the basic concepts of energy,
space, and time. Heightened choreographic imagination—a way of thinking about mass in time and space—can influence how people think through
physics. The skills developed in each field are complementary. Choreographic research strengthens skills of observation, calculation, and problem
solving. Quantitative reasoning cultivates the ability to engage with proportions and relationships, which are central to choreographic thought.
We do not always need to think about physics and dance simultaneously
for the interdisciplinary dialogue to be in process. While the rewards of
making direct ”eureka!” connections are great, most of the time the work of
putting these disciplines together requires going deeply into one field while
pulling the other discipline along. Eventually, the focus will shift the other
way. The inquiry occurs in motion, through a process of constantly changing
perspective. Each discipline gives us lenses through which to view the other.
The goal is to know which lenses to use when, and what to observe, in order
to see something new. This work leads to something more subtle than a
shriek of illumination: it quietly changes how we understand both physics
and dance.
The physical conditions of our universe affect us whole bodily. The same
natural forces that aid in the movements of balls and pulleys act upon us. We
embody space and time, informed by cultural and political forces as much as
by theories of relativity. The physics and dance inquiry in this book conjoins
different ways of knowing our world to help us better understand how and
why we move.


Part I: Principles of Movement

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Our study of gravity begins in motion. Start by lying on your back on the
floor with your legs extended long, slightly wider than hip-width apart. Extend your arms out from your sides at a forty-five-degree angle. As you
lie there, notice the environment you find yourself in. Are you lying on a
soft and malleable surface or on hardwood or cement? Does the floor feel
warm or cold? What sounds do you hear? How far away or nearby are
those sounds? If you are in a building, are you on the ground floor or higher
up, and how does this affect your senses? Notice the inescapable downward
pull on your body.
Gravity glues us to the earth. When you get out of bed, you must push
yourself upright. To walk around, you must push off against the surface of
the planet. Without support—encompassing everything from your legs to
the earth’s crust—you would fall toward the center of the earth. A jump
may give you a short-lived escape from contact with the planet, but even
dancers who jump high cannot jump into space. This is a good thing, since
the air you breathe, subject to the same gravitational attraction, also hugs
the earth. You are attracted to the earth, and the earth is attracted to you,
through the force of gravity.
The art of dance deals with gravity in many different ways. Some dance
techniques are attempts to defy gravity, though its effects are ultimately inescapable. Other techniques opt to accept and give in to its force in order to
produce movement. How we choose to dance with gravity shapes our sense
of self and our worldview. Each dance form in turn expresses a particular
vision for the interaction of humans and natural laws.
In this chapter, we will consider gravity both as a force that shapes human movement and as a natural law that we can model mathematically. We
will help you build basic tools in both disciplines. You can improve your
mathematical abilities with practice. And some of the most basic tools of
dance are available to you simply by moving in the world.

Tuning in to Gravity
Returning to the movement exercise, begin to tune in to the forces that act
upon you. As your body presses downward into the floor, the floor presses
upward against you.


Check in with the various parts of your body. Lift one leg slightly off the
floor. Hold it a moment and then gently set it down again. Lift the other
leg ever so slightly off the floor, hold it there, and set it down. Your legs
may shake slightly as your muscles hold them aloft. Now lift your arms and
head a centimeter off the floor. Suspend them in the air and then lower them
back down. How does the sensation of gravity change when you lift your
limbs? What kinds of muscles must you flex in order to hold your leg or
your arms a few centimeters above the floor? Your task is to remain fully
alert, exploring in this isolated manner the forces exchanged between your
body and the planet.
Now get up—but not as thoughtlessly as you might roll out of bed. Your
movements should be more organized. Set a timer for eight minutes (the
length of time is arbitrary, but setting a time constraint helps structure the
movement). Rise to your feet at a constant speed, filling the entire eight minutes. You may find yourself shifting your weight slowly and purposefully in
reaction to the effort that your muscles and skeletal frame exert to support
you. The key is to cultivate a heightened attention to rising—an action you
have done countless times.
Every so often, pause to monitor the sensation of gravity acting upon
your body. As you work, release any muscles that might be tightening unnecessarily. Relax your head—let it drop toward the floor and sense its full
weight. Relax your arms too, once you have risen to a point at which you
no longer need them to push yourself up. With every pause, scan your muscles and release any unnecessary gripping; this will also permit you to feel
the full weight of your limbs and torso. Feel the direction and quality of
gravity’s downward pull. Work slowly and deliberately.
This movement exercise gives you information about gravity. Through
your actions, you have researched the effect of gravity on the mechanics of
the human body. You have also paid attention to your own psychology and
personal preferences in reaction to the natural force. Memories of prior physical training, old wounds or injuries, or other personal histories may affect
the way you rise. We gain information through movement research: sometimes we can verbalize that information, and sometimes it remains a fleshly
expression of physical and psychic awareness that cannot be articulated in
Gravity is powerful and omnipresent, like an extroverted relative who
has an outsized ability to sway the family dynamic. Human beings recognize the force and calibrate their movements accordingly, because they have
been living with it their entire lives, since floating in the womb. Most people
tend not to shoot up like rockets when rising to their feet: their bodies learn
how much force they must apply and in what direction to overcome gravity
for the desired outcome or result. In dance, this knowledge is one aspect
of muscle memory, the body’s savvy ability to record certain actions under
familiar conditions, including the way such actions feel and how to repeat
As much as dance artists rely on muscle memory to remember dances,

they also look for ways to throw off old habits and reprogram what their
bodies have learned in order to move in new ways. With subtle changes
in approach, such as slowing down a movement, dancers can discover new
information in the most familiar of actions. They defamiliarize everyday
movements in order to transform them into an aesthetic representation, or
You can think of this first movement exercise as a choreographic score, in
which the instructions organize the action. In this case, the score dictated
that you rise up off the floor over a period of eight minutes, while pausing
at your discretion to monitor your body’s reactions. You had a time structure
(eight minutes), a vector (up), and a directive (pause every so often). Add
to these elements all the observations you made while executing this rise:
the details and sensations that entered your awareness and the choices you
made in response. Technique in dance requires sharpening your attention
to the inner workings of your being and your environment, and to their
The movement exercise that you just completed coordinates out of necessity with the physical conditions on our planet. If you were reading this
book on the moon, which has a weaker gravitational field, you would weigh
one-sixth of what you weigh on earth. Your movements would look and
feel very different. But until humans are able to live somewhere other than
the earth, all our dance forms must grapple with the laws of nature on our
home planet, and choreographers must develop an attentive relationship to
the gravitational attraction between the earth and the dancers. That relationship becomes the subject of the dance.

The Universal Law of Gravitation
Why are humans stuck to the planet? What is pulling us down? And how
strong is this force? To understand the effects of gravity on our bodies
through physics, we cannot think of ourselves in isolation. We need to acknowledge the presence of the entire mass of the earth.
As a result of centuries of meticulous observation, scientists have discovered that the gravitational force experienced by two masses depends on the
amount of mass involved and the distance between the masses. The gravitational force that people on earth feel from the planet Jupiter is insignificant—
even though the planet is three hundred times more massive than the earth—
because Jupiter is millions of kilometers away. The gravitational force they
feel from a dancer standing next to them is also insignificant, even though
the person might be separated from them by mere centimeters, because human beings have relatively small masses on the gravitational scale. In order
to experience a gravitational force strong enough to be felt by humans, the
mass must be both large and close. The earth fulfills both of these requirements.
How do gravitational forces behave? We know about forces in the natural world that can repel, pushing objects apart. This is the case for the

electric force felt by two positive charges that are close to each other or two
negative charges that are near each other. We also know about forces that
can attract, like two electric charges that are oppositely charged, giving us a
physics interpretation of the phrase ”opposites attract.” Gravitational forces
are always attractive for a different reason: not because masses are opposite
but because this appears to be the only way gravity functions.
Newton’s Universal Law of Gravitation allows us to calculate the gravitational force (FG ) between ourselves and the earth, or between any two
masses. To do this calculation we will label our two masses m and M. Our
mass is the amount of stuff inside us—the amount of matter. The distance
between the two masses is r. An important point to keep in mind is that the
distance r is not between the surfaces of the two objects but between the center of mass of each object. This makes the distance between us and the earth
not the width of the sole of our shoes but the distance between our center of
mass and the center of the planet, which is approximately 6,500 kilometers
(or 4,000 miles).
In order to figure out the strength of the gravitational force between the
two masses, we also need a constant we label G. If we are measuring mass
in kilograms and distances in meters, G is quite a tiny constant, with a value
of 6.67 x 10−11 N m2 /kg2 , or 0.0000000000667 in units of newtons multiplied
by square meters divided by square kilograms. Newtons (N) are the units of
force. American readers will be more familiar with this unit’s counterpart,
the pound (lb), and 1 N is approximately equal to 0.22 lbs.
The equation for Newton’s Universal Law of Gravitation is:
GM m
We can see from the equation that the force will increase when the masses
get larger. And the force will also increase if we make the distance between
their centers of mass smaller. The gravitational constant has the same value
everywhere in the universe. This number gives us access to an innate characteristic of gravity.
Notice that though two masses are involved, we have calculated only one
force. This is because each of the masses is pulled with the same attractive
gravitational force. When we think about our gravitational attraction to the
earth, we should not sell ourselves short in the relationship. In spite of our
small masses, each of us is an equal partner with the earth. The strength
of the earth’s gravitational pull on us, what we call our weight, is equal to
the pull that we exert on the earth. Because each of us has a relatively small
mass compared to the planet’s, the earth affects our motion, whereas we
hardly affect the earth’s motion at all. But it is perfectly accurate to think of
the earth as our gravitational dance partner.
We can ask another important question: How is it that one mass is aware
of and responds to the existence of another mass that is nearby? Scientists
do not yet know. One theory posits that an as-yet undiscovered particle, the
graviton, carries this information. Scientists still have more to learn about
FG =


how gravity works. Gravity and general relativity are areas of active research, to which we shall return in the chapters on space and time.
With the Universal Law of Gravitation we can see that our mass is gravitationally attracted to every other mass in the universe, and all of those other
masses are gravitationally attracted to us. That we are connected in this way
to every mass in the universe is a little bit thrilling, and pushes the gravitational dance between ourselves and the earth out to a cosmic scale. But how
does gravity influence our movement? How does it affect dance?

Another basic movement exercise will help you deepen your understanding
of gravity. Stand in parallel position: your feet should be approximately one
foot’s width apart and parallel to each other.
Close your eyes. Feel your shoulders over your hips. Turn your attention
to the length of your spine, imagining its trajectory elongating up through
the roof to the sky and down through the floor into the center of the earth.
Relax your neck muscles by gently turning your head right and left. You
have already become acquainted with the force of gravity while both lying
down and rising. Now you are in a new position, standing. Your muscles
and skeletal frame adjust to this new relationship to gravity. Many modern
dance techniques incorporate this parallel position into their warm-ups.
Having focused your awareness on the forces at work in this stance, you
are going to attempt a series of weight shifts. Keeping both feet on the floor
at all times, shift your weight to your right and then return to center. Then
sway to the left and return to center. Try leaning a few inches forward and
backward. Try not to bend forward at the waist—think of yourself as a
tower, adjusting in increments to small plate shifts under the earth. Be sure
not to lock your knees. Unless you are playing a stiff-kneed character—a
sailor in the 1940s musical On the Town, perhaps—no good dancing comes
from locking your limbs.
As you perform this exercise, you will discover that your body makes
minute muscular calculations to keep you from toppling over. The pressure
in your feet will change according to where you distribute your weight. Your
muscles tense accordingly, as well. The downward pull of gravity may seem
to move around as you lean, becoming noticeable on your back, hips, head,
or the side of your body, depending on the direction in which you lean.
Think of your movements not as resisting but rather in conversation with
natural forces.
The positions of the body through which dancers achieve balance reveal
cultural, geographical, and historical attitudes. The exercise you just tried
falls firmly within European-American modern dance traditions. Let’s add
another position to the research, for the subtle differences between dance
forms and cultures can be astonishing seen through the lens of physics.
The idea of sigi, ”sitting,” in the Bamana language of West Africa, lies at
the foundation of many traditional West African dances. The knees are bent,

as if the dancer is about to sit, with the feet still in parallel, approximately
hip-width apart. The torso leans slightly forward, the spine long, as the
pelvis shifts back to accommodate. When you try it, feel the difference between the completely upright torso of your first stance and this tipped pelvis
and more forward torso. Sigi signifies an everyday action, sitting, even as it
enables a readiness for the rhythmic play.1 You may feel a delicate interplay
between how far you tip forward with your torso and the counteraction of
your pelvis, which helps you to maintain balance. The physics of gravity
remains the same, but the dancer’s position changes his or her relationship
to those forces.
Dancers practice balance by following codified rules, which change according to the dance form. Is the torso upright or tilted? Are the arms far
from or close to the body? How to hold the head? Dancers are not limited to
balancing on two feet: sometimes they balance on their hands, head, shoulders, and toes. A spinning dancer faces a different challenge from that of a
dancer who remains statically in place. A dancer spinning on her head has
another challenge yet. And try to picture a dancer spinning on his head on
a tightrope!
Balancing while standing is the first action involved in taking a step.
Walking is nothing more than a series of off-balance forward weight shifts, a
continuous series of falls due to gravity that allow people to move from one
place to another. Try walking around the room. Walk on your heels. Walk
on the balls on your feet. Walk leaning forward. Walk leaning back. Walk
backward, opening your eyes to your environment. Feel the physics?

Center of Mass
The concept of center of mass in classical physics allows us to analyze how
we are able to balance, to predict what will happen when we are off-balance,
and to back up our theories with numbers. Center of mass is not only critical for a quantified understanding of balance—it is also needed to calculate
the gravitational force acting on an object using Newton’s Universal Law of
The simplest object in which to locate the center of mass is one in which
the density (mass per unit volume) is constant throughout—that is, the center of the object itself would be the center of mass. As an example, a sphere’s
center of volume is at the center of the sphere. The same can be said for the
location of the center of mass of a cube that is uniform in density.

Of course, human bodies are neither simple geometric shapes nor uniform in density. In addition, human bodies are not rigid: the relative locations of their limbs, head, and torso can change. At this point we could
throw up our hands (changing our centers of mass) and declare that the calculation of the center of mass of a person is too complicated to pursue. But
fundamental concepts in our dance and physics interaction, like the conditions necessary for a person to balance, depend on the location of the body’s
center of mass.
In order to develop a general formula for the location of a person’s center
of mass, we need to develop a system to model the human body. But first let
us think about a simpler case, in which a set of masses are all arranged on a
line. We will impose an x-axis on this line of masses and designate a positive
and negative direction. We also need to set a point as the origin, where x =
0. We will have positive values stretch out to the right and negative values
stretch out to the left along our line.
The center of mass of a group of masses arranged on this line can be
computed with the following formula:
xCM =

x1 m1 + x2 m2 + ... + xn mn
m1 + m2 + ... + mn


where xCM is the position of the center of mass along our x-axis. The values
x1 , x2 , etc., correspond to the location of the masses m1 , m2 , etc. The formula
to find the location of the center of mass adds a series of multiplications of an
object’s position by its mass in the numerator (the top half of the fraction).

This gives us units of meters times kilograms, if we use the International
System of Units (SI units). The sum of the objects’ masses is the denominator (the bottom half of the fraction). Since the units in the denominator are
kilograms, the final answer will be given in units of meters, and the results
specify the center of mass position along the x-axis.
If we want to calculate the center of mass of a system that has masses in
three dimensions (which is necessary for systems like people, who exist in
three dimensions), we can repeat the calculation for the other two axes that
make up our three-dimensional coordinate system:
yCM =

y1 m1 + y2 m2 + ... + yn mn
m1 + m2 + ... + mn


zCM =

z1 m1 + z2 m2 + ... + zn mn
m1 + m2 + ... + mn


It is convenient that the calculations in the various dimensions can happen independently of each other: if we constricted ourselves to moving
along only one axis we could change the location of our center of mass along
that axis without changing the location of our center of mass in the other dimensions.
Of course, the human body is a mass that is spread out over space, not
chunks of mass condensed at points in space. We can make the calculation
of a person simpler by handling each limb or section of the body independently. If we first calculate the centers of mass of the various parts of our
body, we can then calculate the center of mass of our entire body by treating
each body part as if all of the mass of that body part were located at its own
center of mass. We could therefore have equations that look something like
xCM Lef tArm mLef tArm + xCM Head mHead + xCM T orso mT orso + ...
mLef tArm + mHead + mT orso + ...
We can conceive of our bodies as a system of parts with masses and locations
that can be combined in this formula.
xCM =


How feasible is it to calculate these quantities for a person? Research
can give us the average mass of men and women, and the average sizes and
masses of parts of our bodies. If we are organized and careful, we could
use these data to calculate the center of mass location for an average person, given a specific position. The calculations quickly become laborious,
however, and stay relevant only for the period during which this ”average”
person remains motionless. But understanding how the calculations work
can still give us insight into how our center of mass changes as we move.

Balancing Together
We can now calculate the center of mass of a system: What if that system is
two people joined together as one? Another exercise can help this inquiry.
For this exercise, you will need a partner. Face each other and join hands
by having your partner place her or his hands outward, palms up, while
you lay your palms face down on them. From there, slide your grip up your
partner’s arm, so that you are gently holding each other’s forearms. This
point of contact should feel comfortable—don’t grip too hard. In a moment
you are going to be adding force.
Both of you should place your feet in a wider than parallel position and,
if standing, bend at the knees, being careful to keep your knees in line with
your feet, without rolling inward or outward. The torso is forward: picture
the grounded position of an athlete ready to catch a ball or dart forward or
backward. Your thigh muscles are engaged. You can think of this position as
a deeper than usual knee bend—the stance allows you and your partner to
modulate the tension between you, as if your knees were shock absorbers.
The position allows each of you to sense both gravity and your partner.
Begin to pull away from each other—not too forcibly, just enough so that
you would both be off balance if you let go. Maintain a curve in your lower
backs. Keeping the stretched tension of opposition between you, allow first
one partner and then the other to move in ways that change his or her center
of mass. The other mover must make adjustments to accommodate those
changes to prevent the system you are creating together from tumbling in
one direction or another. One person explores the possibilities of shifting
the position of his or her arms, feet, and legs, while the other must maintain
the system’s balance by shifting forward or back, deepening the bend in the
knees, and so on. Take turns leading and accommodating the changes in
center of mass. Once you feel comfortable with this first effort, explore different configurations, such as standing side by side and pulling away from
each other holding on to only one arm. Try not to speak to your partner;
instead communicate through physical negotiation.
This exercise allows you to research the conditions for balance between
two systems that become one. People are different from inanimate objects, of
course, and your movement research must also take into account the physical and social negotiation between you and your partner as one aspect of
the information gained. But when we model this exercise using physics, we

temporarily set aside the fact of social interaction in order to focus on the
interaction between your bodies and the earth.

Area of Support
When your center of mass—or that of the system consisting of you and your
partner—extends beyond your base of support, you fall. But how can that
limitation be quantified? If you stand with your feet directly underneath
your hips about a foot’s width apart, how large is the area of support?
If your feet are 25 centimeters long—a U.S. size 7 shoe for women or
about a size 6 shoe for men—and a maximum width of 10 cm, we can make
an approximation of the area of support as a rectangle that is 25 cm long and
30 cm wide if you stand with your feet in parallel. This gives a total area of
25 cm × 30 cm = 750 cm2
If you stand with your center of mass directly over the center of area
of support, you can lean 15 cm to the left and right, and 12.5 cm forward
and backward before you begin to fall. If, however, you are working with a
partner, clasping each other’s forearms with your centers of mass separated
by one meter when you both stand upright, the area of the base of support
has been extended to roughly 1.250 m in length and 30 cm in width. Your
combined area of support is then
1250 cm × 30 cm = 37, 500 cm2
Together, you and your partner are able to establish a larger area of support than either of you could individually. Assuming that your partner ac17

commodates you, this allows you to lean considerably farther forward, backward, or sideways without losing your balance. Other sculptural forms that
you and your partner might imagine might be possible to enact if you add a
little physics analysis to aid in your discovery.

The Dance of the Future
The force of gravity on earth has remained constant for billions of years. But
if we turn our lens from physics back to dance once again, we find that the
nature of gravity—or more accurately, the dancer’s relationship to gravity—
has been more contested. In the development of American modern dance
in the first half of the twentieth century, for example, gravity figured into
polemical debates about power onstage.
In 1903, the dancer and choreographer Isadora Duncan gave a lecture
that she later published as a manifesto titled ”The Dance of the Future.”
Duncan’s dance of the future stripped away what she saw as the artifice of
classical ballet by tapping back into natural forces. As the cornerstone of her
vision, Duncan argued for a new relationship to gravity:
The dance should simply be, then, the natural gravitation of this will
translation of the gravitation of the universe.2
In Duncan’s view, the individual’s will—his or her very impulse for moving—should align with the universal laws of gravitation. By working with,
rather than against, gravity, her dance would channel a powerful cosmic
Large ballet companies led by male directors dominated the Western concert dance world at the turn of the twentieth century. Duncan’s manifesto
was a direct refutation of classical ballet, whose trappings—from the training to the costumes to the patriarchal culture—she felt constrained dancers,
especially ballerinas.
Her revolution played out in technical terms: while classical ballet technique asked dancers to pull up, Duncan’s modern dance insisted that they
pull down—toward the earth, rather than away from it. Being physicists
as well as dance artists, we know that we cannot will our way into a new
gravitational field—both ballet and modern dance ultimately deal with the
same physical conditions. And the opposition that Duncan envisioned has
become much less sharply defined in the one hundred years since she published her manifesto: ballet technique is more grounded and contemporary
dance is more ethereal than she could have imagined. But the conceptual
split that she identified between ballet and modern dance, based on the relationship of each to gravity, persists to this day.
With the deceptively simple act of performing earthbound movements
that she had created herself, among other strategies, Duncan broke free of
the restrictions imposed on women’s bodies in performance. Gravity aided

her symbolically, as much as physically, to tip the art of dance into a new
sociopolitical age. For her innovations, historians have identified Duncan as
a mother of modern dance.
Fast-forward fifty years to consider the work of the dancer-choreographer
Pearl Primus, in which the dancer’s relationship to gravity took on another
political dimension. Primus was an anthropologist who incorporated her
fieldwork into her dances. She was one in a line of anthropologically trained
choreographers, including Zora Neale Hurston and Katherine Dunham, who
shared an interest in synthesizing African diasporic forms with EuropeanAmerican modern dance. Offering another vision for the dance of the future
that promoted Africanist cultures and aesthetics, these artists fought racism
and championed the complexities of African American culture and history.4
Primus injected into her dances rhythm, fluidity, daringly high jumps,
and fearless falls. Like Duncan, she danced with—not against—gravity, but
in her case the gesture tied her to the dances that she had researched in the
Gold Coast, Angola, Liberia, Senegal, and the Belgian Congo, among other
countries on the African continent of the late 1940s.5 She describes the earth
in African dance as ”an extension of the dancer’s own feet, as if it were a
stage of rubber from which he can bounce to the skies, as if it were a soft
bed upon which he could roll and be protected.”6 Primus incorporated this
Africanist vision of oneness with the earth into her choreography.
Study Primus performing her solo Spirituals in 1950 to witness the power
of such a vision.7 Primus leaps, legs in a V, arms open to the sky, then drops
to the ground to execute a breathtaking series of forward rolls. Moving in
a tight circle, she falls face-first onto the stage floor, rolls, and then quickly
pushes herself back up to her knees—her center of mass low to the ground
and her chest high, as if ascending from the deep. She does this over and
over again. It’s an image of redemption expressed in movement, and gravity
is her guide.

Interdisciplinary Thinking
As you can see, physics and dance have different ways of understanding
gravity and center of mass, and they do not always neatly coincide. In fact,
their juxtaposition can be awkward, as we move from exploratory movement exercise and qualitative research to mathematical problem solving.
This awkwardness is intrinsic to interdisciplinary thinking; it is also the
source of interdisciplinary discovery. Remarkably, two bodies together, without using words or numbers, can figure out how to create a system that remains balanced. Also remarkable is that, armed with Newton’s Universal
Law of Gravitation, we can use the motion of our bodies to calculate the
mass of the earth.
In this chapter you had an encounter with the process of synthesizing
different disciplinary methodologies and ways of knowing to gain deeper
understanding of each. As we proceed, we will ask you to keep an eye on
questions of comparative knowledge acquisition: How is knowledge gener19

ated, what forms does it assume, and what can the combination of different
ways of knowing tell us that we could not have found out through one discipline alone?
We are juxtaposing two different ways of understanding the forces of nature that affect us all. Physics gives us a framework within which to analyze
and predict motion. Dance structures the perceptual experience that helps
us understand the effects of this motion in our lives. The human body in
interaction with the natural world is a highly complex system. As we have
seen, we can heighten our awareness of natural forces through movement
practices, even as we learn how to describe how these forces work using
numbers on the page.




The ”duet with gravity” that you created through the previous chapter’s exercises offered you a focused way of relating to natural forces. By slowly
ascending from the ground, you acquired greater knowledge of the sensation of gravity upon your body. You also built knowledge of some of the
ways that human anatomy can both resist and comply with that force. You
experienced giving in to gravity and permitted the force to motivate your
actions and shape your physical structure. You also cultivated an awareness
of basic conditions for balance by creating positions—in physics terms, distributions of your mass—that helped prevent you from falling over. Dance
builds upon these basic movement research experiments. Think of dance as
a three-way interaction among natural forces, our awareness of those forces,
and our physical imagination and capabilities in response to them.
You may have found that any previous physical training you have had—
from flamenco to football—informed your experience of the gravity exercise. The information you acquire from practicing these physical techniques
is one aspect of your embodied cultural knowledge, which is another form
of ”force” acting upon your body. Gravity is gravity, on earth, Mars, and
beyond—but our physical, kinesthetic, and psychological relationships to
these natural forces change according to the dance form we choose, and
thereby the quality of the movement also changes.
Both natural and cultural forces shape how we move: the question is
how to make those forces visible. Every dance practice embodies an entire
cultural belief system—a cosmology of action and reaction in relation to the
natural world. Physics probes those natural forces through theories, formulas, and diagrams. If we pull the scientific and cultural views together, we
might be able to construct a more complete picture of human existence.
How do we measure motion? And what drives any form of human
movement? The answer is force.

Newton’s 1st Law of Motion
A dancer sitting on the floor must do some work to start moving and would
need to do more to keep moving. This may make us think that our natural
state is to be at rest. However, as we will see from Newton’s 1st Law of
Motion, the natural state of an object is more complex: it takes force to slow


us down or speed us up, but not to keep us moving. Newton’s 1st Law of
Motion states:
A body at rest stays at rest and a body in motion stays in motion unless acted on by
a net external force.
Forces are pushes or pulls. They are vector quantities: they have an amount—
or magnitude—associated with them as well as a direction in which they are
applied. To calculate our net, or total, external force, we need to add up all of
the forces acting on us while taking into account the direction in which they
are acting. Imagine you are standing between two people, with one person
pushing you to the left and the other pushing you to the right. In order to
calculate the total force acting on you, you need to take into account that
these two forces are working against each other. A force of 10 newtons (N)
to the left plus a force of 8 N to the right equals a force of 2 N to the left.
It might not seem to be a natural state of things for an object in motion to
stay in motion because every time we start moving we have to work to keep
moving, due to gravity and friction. One of the difficulties in understanding
Newton’s 1st Law is that we constantly interact with forces as we move. You
may find that you need to change your intuitive sense of motion to conform
to Newton’s Laws.
As a start, imagine a place with fewer forces at play—outer space, for
example. (You will realize as you read this book that physicists often send
dancers into space. This is because the environment on the surface of the
earth is much more complicated than the one in space due to the gravitational pull of the earth and frictional forces on earth’s surface.)
Let’s put you in a comfortable space suit with plenty of oxygen very far
away from large masses, so that the force on you due to gravity is negligible.
How would you begin to move from your position in space? You don’t have
many options. You need some kind of push or pull on your body in order
to get started. And once you began moving in deep space there would be
nothing to slow you down or stop your motion: no friction, no air resistance.
In this context it seems perfectly clear that your natural state is to continue
as you have been going, either at rest or in motion, unless you encounter
external interference. This is exactly what is stated in Newton’s 1st Law of
Is this also the natural state of things on the surface of the earth, where
both gravity and friction come into play?
Let’s continue our thought experiment and introduce a meaningful force
of gravity but keep the forces due to friction minimal. We will put you in
a skating rink with such smooth and slippery ice that you cannot get any
traction. Gravity pulls you toward the center of the earth, but that force
is exactly balanced by the ice pushing up on you, keeping you at a fixed
distance from the earth’s core. If you are at rest, you cannot start yourself
moving. Your feet would just slip if you tried to run on the ice. Conversely,
if some push or pull were to get you in motion, you would keep going (for

a while) since there would be only the minimal force due to friction to slow
you down. Again, these results are consistent with Newton’s 1st Law.
Now try walking in socks on a dance floor (or in sneakers on the sidewalk). Frictional forces will require you to keep putting energy into maintaining your motion. This experience is what makes Newton’s 1st Law of
Motion so difficult to understand—it appears, due to these external forces,
that our natural state is to come to rest. But remember, the continued effort
to keep moving was not necessary in the ice rink or in outer space. External
conditions can create frictional forces that interrupt an object’s motion.
But before you conclude that friction acts only as an impediment to motion—it slows you down and requires you to work—keep in mind that it
also gives you tremendous control over the direction you go and the speed
of your motion. We will take a detailed look at friction in Chapter 4.

Newton’s 2nd Law of Motion
Newton’s 1st Law of Motion is really a subset of the 2nd Law. The 1st Law
tells us what happens when there is no total force acting on an object—
namely, that it will stay as it is, either in motion or at rest, whichever it has
been. The 2nd Law allows us to calculate how the motion changes through
acceleration due to the external forces. The law can be stated, first in words
and then as a formula, this way:
The sum of external forces acting on an object equals the object’s mass times its
(F ) = ma


where the symbol
is a summation sign.
On the left side of the equation, the forces are added up. We must keep
in mind that we need to keep track of their amounts as well as their directions because they are vector quantities. On the right side of the equation,
the mass of an object m is multiplied by acceleration a, which is also a vector quantity. Imagine what happens to the acceleration when the amount
of force changes but the mass remains unchanged. As the force increases,
so will the acceleration. If instead the force stays the same but the mass increases, the resulting acceleration will be reduced.
This equation makes sense because it is more difficult to get an object
with a lot of mass moving than an object without much mass. As m goes up,
the value of a will decrease for a certain net force. It is also more difficult to
slow down an object with a lot of mass that is in motion than it is a lowermass object. This equation is incredibly useful because it lets us quantify the
acceleration of an object under any set of forces.


In focusing on acceleration, Newton’s 2nd Law of Motion can give us
insight into one of the most universal dance movements: bending at the
knees, which is often referred to as a plié.

The Plié
Bend your knees while standing, and you will appear to get shorter. But being ”shorter” does not imply that you are less powerful: in fact, this movement will prepare you for any movement that might come next. Aided by
your coordination, energy, and the type of movement that came before, a
deceptively simple knee bend helps you muster the resources you need to
dance. If we model this movement in physics terms, you first accelerate
downward, and then you accelerate upward. The amount of the acceleration will differ, depending on the dance technique.
In classical ballet technique, this action is called a plié, a French word that
means ”bend” or ”fold.” The term has carried over to certain modern dance
forms as well. Bending the knees lowers the body’s center of mass, drops
the pelvis, and allows for a more perceptible relationship to gravity. The plié
allows for greater reactivity: a deep plié can mean the difference between
having more force available to rev up and get going and trying to leap from
a lock-kneed standstill.
The means by which a dancer accomplishes this simple knee bend vary
dynamically depending on the dance technique. In certain West African
dance forms, the knees bend swiftly, dropping the body’s center of mass
quickly. This plié acts less as a cushion than as a propulsive action through
which to connect with the earth. From this lowered position, the dancer has
control over rising, rhythms, and more. In contrast, in Bharatanatyam, one
of the oldest forms of Indian classical dance, dancers frequently maintain a
bend in the knees with the feet turned out. From this position, known as
ardhamandala, the dancer drives the form’s characteristic rhythmic footwork
down into the floor. These various schools of thought express an entire ethics
of how to move through, and relate to, the world.
Different schools of thought concerning the plié exist even within the
same dance form—and individual teachers of the form sometimes differ the
most. The Russian ballet choreographer George Balanchine, a founder of
New York City Ballet, asked his dancers to accent the start of the plié and
then slow the movement, as if applying shock absorbers. After reaching the
bottom of the plié, which is very deep, the dancer would quickly return to a
standing position, drawing the legs back together. The first count of the plié
is thus a kind of bottomless resource, while the second count either returns
the dancer to the starting position or results in surprising action: a pirouette,
a jump in the air, a rise onto pointe.
Stanley Williams, a legendary teacher at Balanchine’s School of American Ballet, took yet another approach. He used enigmatic imagery: ”And
you’re in,” he would say of a plié. Instead of thinking of the plié as a string
of discrete opposites—moving the body down and up, bending the knees

out and in—Williams wanted circularity. Pulling up was part of lowering
down, and pressing down was part of rising up. Likewise, the knees bending outward already contained the ”in” of their straightening. Classical ballet comes with a set vocabulary of positions, which have been passed down
and transformed from the fifteenth- and sixteenth-century European aristocracy to the present. Ultimately, Williams wanted to see no extraneous actions. His plié helped dancers transition seamlessly between positions, without the fidgeting that would have interfered with pure ballet form. Those
who danced under Williams’s tutelage felt that his plié possessed a kind of
existential truth, and spent many hours attempting to perform his vision.
For our purposes, Stanley Williams’s ”in” might also be understood as
”in the direction of gravity’s pull,” or inward toward the center of the planet.
His vision of plié kept dancers from skittering on the surface. As with
Bharatanatyam, West African dance, and many other dance forms, the plié
and its variations are fundamental to dancing because all dancers need to
figure out how to relate firmly to the ground upon which they move.

Down and Up
We left you with the equation (F ) = ma and then veered off into the philosophy of bending at the knees and how various schools of thought in dance
forge relationships to the earth. How do these relate to physics? In fact, we
never left the physics behind, for every time you plié you are engaging with
the phenomena described
P by Newton’s 2nd Law of Motion.
To understand how (F ) = ma is at work in the simple action of bending your knees, we will ask you to perform three variations on executing a
plié in time. Keep in mind that with each version of the plié, you are playing with different accelerations along the vertical axis—Newton’s 2nd Law
should be whispering in your ear.
Stand with your feet in parallel position, one foot’s width apart. Recall
that your shoulders should be over your hips and your knees over your
toes. Feel your spine elongate directly upward, as if hanging from hooks in
the ceiling—a frequently used image in ballet and modern dance training.
Be sure not to lock back in your legs: maintain a gentle give in the knees.


The ability to sense and react to the forces acting upon your body is largely
contingent on your interaction with the force of gravity.
Now, we are going to organize your plié in time. You may recall that in
the previous chapter we structured your rise from the floor in a unit of eight
minutes at a constant rate. In this movement exercise, we are going to finetune that structure further and set the movement to beats, defined in dance
by repeating, regular intervals of time. (Note that this is not how ”beats”
is typically defined in physics, where the word refers to the interaction between two frequencies, as occurs in tuning a musical instrument.) The beats
can be slow or fast at your discretion, but the important thing to remember is that the beats give your actions a temporal framework—think of it as
a support system for your movements that helps give them greater shape.
Beats are the rudimentary building blocks of music, to which so much dance
In this exercise a metronome will be helpful. (Metronome apps are readily available online.) With your metronome ticking away, take two beats to
bend your knees and two beats to rise up again. Be sure your knees are
aligned over your toes when you bend, and your shoulders remain over
your hips. Your head is held upright and your gaze steadily focuses straight
This is a plié in parallel, one of the most basic positions in modern dance
technique. The evenness of these counts—two to go down, two to come
up—have a metric regularity, like well-oiled pistons in a machine.
Now, for the second exercise, angle your toes slightly outward, keeping
your heels together and your knees over your toes. This is first position in
classical ballet. From this position, try a ”demi”-plié, which means going
as low as you can go while still keeping your heels on the floor, your spine
elongated, and your pelvis tucked under. Use a different timing for this plié:
accent the start of your descent on the first beat (loosely put, think of the
accent as a ”start quick then go slow” timing). By the end of the first beat,


reach the full bottom of your plié. Then return to straight legs at the last
second, by the end of the second beat. Try this a few times, to get the feel of
Finally, in this same turned-out position, try a third approach. Go to
the bottom of your plié, maintain the bend in the knees and the turnout,
and move forward around the space in which you are working. Advance
heel first—that is, your heels should make contact with the floor first. Try
changing directions, keeping the bend in your knees as you travel. Your
thighs may burn, but no one said dancing was easy.
A good teacher will teach plié not as a rote exercise but as an act of discovery: of music and forces, self and world. Lessons on how to plié are
best transmitted body to body by a live dance instructor, or even several
live dance instructors, for each technique is a detailed language that dancers
take years to master. To be sure, so much information is packed into any one
dance step that a holistic description of the movement would far exceed the
limits of the written page.
By performing three different types of plié, you have been researching
how different technical approaches in dance operate and feel. You have
also been researching Newton’s 2nd Law of Motion, for the many technical
details—such as timing, rhythm, and intention—lead to a unique acceleration in relation to gravity. We can deepen your research into the degrees and
directions of the forces involved using what are called free body diagrams
and force plates.

Free Body Diagrams
In order to analyze the three variations on a plié using Newton’s laws, you
can create a free body diagram. To make a free body diagram, first draw a
representation of the object you are analyzing. Then indicate with arrows all
the forces that act on the object. These are the forces you need to keep track
of in order to do a calculation with Newton’s 2nd Law. In this example, you
are going to analyze a dancer doing a plié.
Keep in mind the distinction between the forces that your body is exerting and the forces that are acting on you. The forces your body is exerting
include the force of your feet pressing down on the ground and that of your
body pulling the earth toward it through gravitational attraction. The forces
acting on you include the gravitational pull of the earth on your center of
mass and the floor pushing up on your feet. Since your acceleration is determined only by the forces acting on you, include only those forces in the
Draw an arrow that indicates the direction in which each force is acting,
and indicate the magnitude of the force, either with a numerical value or
with a uniquely defined variable. Once you have included all forces acting
on the body in your free body diagram, you can apply Newton’s 2nd Law
to calculate the resulting acceleration by adding the forces as vector quantities. In the sample free body diagram of a dancer in a plié that follows, we

indicate the force due to gravity with the label FG and the force due to the
ground with the label FN .
A free body diagram is a powerful tool for analyzing the forces of nature
acting upon a body. At the same time, the diagram has its limitations. It does
not communicate history, for instance: it is a freeze-frame study, capturing
the state of a body at one instant in time, without information about the motion that preceded that instant. A free body diagram of the body in midair,
for example, would look the same regardless of whether the body was on its
way up, at the height of a jump, or on its way down. In each instance, only
the force due to gravity is at work. You can use the diagram to calculate the
acceleration at that instant, which tells you how the object’s current velocity
is about to change. But the calculation will not tell you how fast the object is
currently moving.
For the plié, a free body diagram of the moment when you begin your acceleration down will show a greater force due to gravity than that from the
floor. In contrast, at the point when you begin to accelerate up a free body
diagram will show that the stronger force acting on you comes from the
ground. During the moments of constant velocity, whether you are moving
up or down, the forces are perfectly balanced and there will be no acceleration.
A free body diagram leaves out other information as well, namely, the
cultural contexts that inform human movement. Highlighting cultural forces
within dance requires different modeling systems and theoretical frameworks, such as those found in the social sciences, humanities, and dance

A Cultural Force Diagram
We used the free body diagram to map out the natural forces acting upon the
human body in specific instances during a plié. What happens if we borrow
this idea for the purposes of analyzing the cultural forces at work? We use


the idea of ”culture” broadly here to describe both the codified rules inherent
within a given dance form and the dancer’s own personal movement history
as it channels the social and political environments in which he or she lives.
Create a new diagram of your plié, using any of the variations on the
plié that we have discussed. This time note any influences of movement
training that you perceive to have had an effect on your experience of the
plié. Do your knees long to turn in or turn out? Do you prefer to hold your
spine elongated upright, angled, or stooped in relation to the floor? Why do
your feet behave the way they do when you walk? Why do you hold your
head steadily upright on your spine, or allow it to drop and react to gravity?
What can you learn from how you hold your hands? Depending upon your
prior physical training and social conditioning you may experience gaps—
conflict, even—between the details of the dance technique and your body’s
preferred manner of moving.
Try to identify and spotlight details in the movement that relate to your
own personal background and training. A wide range of physical training
and environmental influences can count—from the types of spaces you usually occupy to your own internal sense of time, which is often culturally
informed. (You might even include ”impatience” as a cultural force inherited from the age of social media!) The goal is to unearth the movement
cultures that influence how you move, for how you move shows you something about who you are.

Force Plates and the Plié
With the free body diagram and the cultural forces diagram, you have sketch-


ed out different forms of force acting on a dance movement. You can take
your analysis further by adding a quantitative layer with the help of a force
plate. A force plate enables you to measure forces that act on the plate’s surface over time. Like a scale, a force plate continuously reads the force that
presses upon it, but it is more precise in its readings than a typical bathroom
scale, and it allows you to store the output on a computer for later analysis.
If you stand still on a force plate, it will read your weight, which is your
mass times the acceleration of gravity, since gravity is the force that is acting
on you.
(F ) = ma = mg
where g is the acceleration due to gravity at the surface of the earth.
The reason you are not accelerating due to gravity as you stand still is
that the floor is pushing back up at you with an equal and opposite force.
As long as your center of mass has no motion with respect to the floor, the
forces are equally balanced. How, then, do you lower your center of mass,
as you do when you bend your knees in a plié? As you start to bend your
knees, the force that the ground can exert on you is reduced. At least for a
time, you will accelerate toward the floor.
Different variations on the plié will cause different accelerations and result in different force plate responses. You can imagine quickly accelerating
to a constant velocity that you maintain for most of your plié. During the
times when your body is moving with constant velocity, the forces of gravity pulling down on your body and the floor pushing up on your body are
perfectly matched. This can occur even when you are moving toward the
ground, as long as that motion occurs with zero acceleration and therefore
constant velocity. When, however, you are changing your velocity, the forces
acting on you are unbalanced, resulting in a non-zero net force and therefore
an acceleration that can be calculated with Newton’s 2nd Law. It is particularly instructive to try various plié techniques on a force plate so that you
can clearly see the times when the acceleration is occurring and can map this
information onto your motion. If you do not have access to a force plate, try
the exercise on a bathroom scale. It will not be as sensitive as a force plate,
but it should read something other than your weight when you are accelerating: a lower value than your weight when you are accelerating toward the
floor and a higher value than your weight when you accelerate upward.
An example of what a plié might look like on a force plate is shown in the
drawing that follows, labeled to indicate various components of the motion:


a : The dancer begins standing on the force plate, which reads his or her
b : To begin the plié, the dancer starts to bend his or her knees and accelerate down.
c : There is a period of time when the dancer’s center of mass descends
with a constant velocity toward the floor.
d : At the bottom of the plié the dancer stops the downward motion and
begins to go back up, which is a period of acceleration.
e : The dancer applies more force than needed to change the direction of
motion. (This stage does not always occur.)
f : The dancer’s center of mass moves at constant velocity upward, away
from the floor.

Newton’s 3rd Law of Motion
There is one more law of motion that will help to illuminate the forces at
work in pushing up from a plié, as well as many other movements that a
dancer undertakes: Newton’s 3rd Law of Motion.
This final law refers to two objects exerting forces on each other, whether
planets interacting through gravity or dancers moving in contact. This law
can also explain the interaction between your feet and the floor, or your hand
pressing on the wall of a dance studio or a chalkboard. The expression for
Newton’s 3rd Law of Motion, when objects A and B interact, is:
FAonB = −FBonA


where FAonB represents the force that object A exerts on object B and FBonA
represents the force that object B exerts on object A. The negative sign means
that one force is acting in the opposite direction from the other. (Remember,

forces are vectors, so they have both an amount and a direction.) Newton’s
3rd Law tells us that these two forces are equal in magnitude and opposite
in direction.
We have seen an example of equal and opposite forces with the Universal
Law of Gravitation. People exert the same amount force on the earth as it
exerts on them. The earth pulls people toward its center and people pull
the earth toward their centers—the forces are in opposite directions. Objects
interacting through gravitational forces are one example of Newton’s 3rd
Law behavior.
Initially, Newton’s 3rd Law can be perplexing when it is applied to objects in contact. When you press your hand against a wall, the 3rd Law
says that the wall ”responds” with an equal and opposite force. Change the
amount of pressure you apply to the wall, and the wall changes the amount
of pressure it applies to you. How could the wall know how much force you
are applying and respond immediately and appropriately? Where do these
equal and opposite forces come from, and what constrains them to Newton’s
3rd Law behavior?
When we think of dancers moving in the studio, we must remember that
gravity is not the only source of force between objects. We need to also take
into account the electrical forces that are involved between the atoms that
make up the dancers and the floor. With regard to electrical forces, it is
helpful to remember the common saying ”opposites attract”: two opposite
charges attract each other. The strength of the force each exerts depends, like
gravity, on how far apart they are from each other. But while the strength of
gravity also depends upon the masses involved, the strength of the electrical
force depends upon the charges involved. A positive charge and a negative
charge will be attracted to each other with equal and opposite forces. When
two positive charges or two negative charges move toward each other, they
will repel each other, resulting in equal and opposite forces. In both scenarios Newton’s 3rd Law applies.
How can we understand these forces when two objects are making contact, such as your hand against the wall or your feet on the floor? Although
the details are more complicated than described here, imagine the surfaces
of everything—you, the floor, the wall, your shoes—covered in springs, as
a metaphor for the electrical forces involved. The atoms that make up your
hand, the wall, and all matter have electrical charges in them. When you
bring two atoms closer together, the interacting charges will make the surfaces repel each other. As you move two surfaces closer together, it is as if
you were compressing springs, giving equal and opposite forces back on the
objects in contact. When you push into the wall with your hand, the wall
pushes back with equal and opposite force, giving you Newton’s 3rd Law
behavior because of the spring-like nature of the electrical forces involved.
Even the gentlest contact with another dancer’s fingertips will give the equal
and opposite forces of Newton’s 3rd Law.


How Do We Move?
A common confusion can arise with regard to Newton’s 3rd Law: If every
force has an equal and opposite partner, how can anything ever move? Push
on a box, and the box pushes back equally. How do you slide it across the
room? The answer is that only net external forces are counted when determining an object’s motion. To calculate what a dancer’s acceleration across a
studio will be, we only need to know the forces that are acting on the dancer,
not the forces that that dancer exerts on other surfaces. A free body diagram,
which includes only the forces acting on the body in question, will help with
this calculation.
If you are standing in the dance studio and you would like to begin to
move, how would you accomplish this? As you stand on a floor you have
at least two forces acting on you: the force of gravity due to the interaction
between your mass and the earth’s, and the force of the floor on you. If
you stand at rest, with zero acceleration, these forces are perfectly balanced.
The ”springs” on the bottom of your feet and the surface of the floor have
So how do you get the unbalanced force you need in order to begin to
move? One option would be to have someone give you a push (carefully).
That could start you moving. But what if you wanted to start moving on
your own?
All you need is contact with something. In this instance, you are in contact with the floor. If you push on the floor, the floor has no choice but to
push back on you with an equal and opposite force. If you were to draw a
free body diagram of this scenario, keeping track only of the forces acting on
you, you could see how your pushing resulted in a net force that would drive
you in a direction opposite to the one you applied to the floor. If you want
to move forward, push backward against the floor. If you want to move to
the left, push to the right; and so on. The more force that you successfully
apply, the greater your acceleration in the opposite direction.
In one simple formula, Newton’s 3rd Law of Motion illuminates the mechanics of human movement. A surface you push against actually pushes
back on you, which makes possible a full spectrum of dance forms. Think
of this transaction as a continuous dance with your environment: you are
never dancing alone.

Everyday Action
The relationship between forces described in Newton’s 3rd Law enables us
to perform all of our daily actions—sitting, standing, walking, running, skipping, jumping. Whereas physicists can predict the average forces involved
in each of these actions, suggesting a certain consistency in their value and
interpretation, how a choreographer frames those movements can alter their
meaning entirely.
Consider the action of scrubbing the kitchen floor on hands and knees,


which the dancer-choreographer Blondell Cummings incorporated into her
seminal 1981 solo, Chicken Soup. As Cummings choreographed the dance,
the performer alternates between repetitive, rhythmic scrubbing—her long
arms pushing the brush away and back toward her body, away and return—
and up-on-her-feet dancing, which is wiggly, energetic, seemingly unbounded. Cummings’s composition alternates formally between the recognizable
action of scrubbing and the more ambiguous meanings of her upright dance.
As the solo progresses, she dance-ifies the movements involved in cooking, too: through rhythm, repetition, and expansive execution, they become
choreographic. The set minimalistically evokes a kitchen, with a table and
chair. Cummings wanted to create a solo about women and food, which
spoke not only of her own African American family background but to a
universal domestic experience for women across cultures.
What are the forces involved in Cummings’s movements? From the perspective of physics, we can think about her actions quantitatively. We know
from Newton’s 1st Law of Motion that she needed a force to get going. There
are biomechanics involved here—food becomes energy for the muscles that
press into the floor to create Cummings’s rocking, scrubbing motion. Newton’s 3rd Law allows us to calculate the force she exerts into the floor and the
force the floor exerts back on her. We could also measure the angle of those
forces, as we shall see in Chapter 4.
Apply Newton’s 2nd Law to calculate the force of her arm muscle and the
force from the floor upon the scrub brush, and you can calculate the brush’s
acceleration. However, as you might notice should you do a comparative
study of floor scrubbing across contexts, her force and engagement with the
brush possess an unusually dynamic intensity, for she is dancing the action.
A snapshot F = ma would not necessarily pick up on the waxes and wanes in
force that constitute the rhythmical structure by which she turns her scrubbing into dance.
Cultural forces are involved, as well. This is a dance about the socially
determined gender roles prescribed throughout American history, represented in the domestic space of a kitchen and Cummings’s tasks. Also salient
are the critical interpretations that viewers confer upon race. Although Cummings intended the dance as a universal statement about womanhood, because she was a black woman the solo has frequently been interpreted as a
black protest piece.

When Is Walking More Than Walking?
Everyday action in concert dance takes on different meanings depending
on the artist, time period, and audience. Blondell Cummings worked in an
aesthetic lineage that reached back to the early 1900s, when Isadora Duncan was reacting sharply against what she believed was the harsh artifice
of classical ballet training on the body and designing a way of dancing she
deemed more natural. Duncan’s movement was not totally extracted from
precivilized existence—her aesthetic influences included the philosophy of

Nietzsche and classical Greek sculpture. But she nonetheless opened the
floodgates to a new way of dancing before the public—solo, as a white
woman—partly by including movements that many in the audience could
have performed.
In Weimar Germany in the 1920s, the choreographer and theorist Rudolf
von Laban not only incorporated pedestrian movements, he also opened
dance to the untrained when he created massive movement choirs with hundreds of participants. Everyday action became large-scale dance spectacle in
his hands.
In New York City during the 1960s, an adventurous group of young
choreographers pushed the type and range of actions that could be included
in dance farther, partly by inventing new choreographic structures to organize those movements. They created rigorous spatial and temporal scores
to contain everything from walking and running to moving mattresses and
eating apples. The group presented their work under the collective Judson
Dance Theater and have since come to be known as pioneers of postmodern
Cummings emerged as a performer in the late 1960s out of Judson circles of influence. She developed her own unique spin on composing with
everyday action: whereas Judson artists decontextualized everyday behaviors in their compositions, Cummings returned them to their original context
by alluding to their settings—as with the spare set that evoked a familiar
kitchen—thereby charging those actions with a social commentary all her
Using pedestrian movements, dance artists have challenged classical ballet’s elitist domination in dance, brought high art to the masses, democra-


tized the space of public performance, and drawn attention to marginalized
experiences. When the movements onstage look more like everyday life than
virtuosic technical feats, hierarchies begin to topple.
If walking is presented as walking—without expressive flair or kinesthetic flourish—what pushes it into the realm of art? One way to answer
this question is to consider the artist’s compositional choices: the temporal structure, spatial organization, and movement vocabulary all impact the
viewer’s perception of any movement, whether spectacular or quotidian. By
subtly arranging a walking pattern, or juxtaposing movements against each
other—the action of scrubbing against interpretive dancing, for instance, in
the case of Cummings’s formal innovation—the choreographer can render
something that is familiar unfamiliar. Through this defamiliarization, we
can be jolted into seeing our world afresh. To quote the choreographer
Yvonne Rainer quoting the composer John Cage (who was himself repurposing a verse in the book of Ecclesiastes), there is nothing new under the
sun; there are only new ways of organizing it.8




Most people get through the day with a very narrow range of foot activities:
their feet usually assist with standing and transporting them from here to
there, often by walking or running. But feet have so much more to say than
this limited repertoire suggests. Consider the many different ways your feet
can move. In dance, feet can slide, slap, tap, stomp, flex, flick, relax, and
point. They can help the dancer leap upward and outward, and then cushion
the descent when he or she lands.
When it comes to leaping in physics, Newton’s 3rd Law of Motion establishes that as people push off the planet to get moving, the planet pushes
back at them. If they manage to push hard enough, the planet can launch
them into the air—at least for a brief period of time. A new set of equations
enables us to access the symmetry that lurks in the phenomenon of jumping.
Combined with an investigation into the nitty-gritty details of how dancers
jump, these equations can help illuminate how dance achieves its expressive
Physicists typically set up hypothetical situations with objects to explain
how to calculate the variables involved in projectile motion. In this chapter, we replace scenarios that use inanimate objects with dancers and their
physical techniques. One way to understand physics concepts is through the
motion of the human body—and few people have greater agency over the
physical forces acting on their bodies than dancers. A dancer’s knowledge
will affect the quality, height, and timing of the jump, even as the physics
tells us how high, how far, and how long the dancer can manage to fly.

Dancing Feet
Dancers use their feet in specialized ways: to slam down or roll through, to
tread softly or consume space, to produce musicality and rhythm, to convey
character. In dance, feet speak.
Acquiring sophisticated knowledge of footwork takes practice over time.
Depending on the dance form, a dancer’s feet develop different habits. Ballet dancers tend to point their toes any time their feet leave the floor. Other
dancers use what we might call a ”relaxed point,” which is somewhere between pointing and actively flexing at the ankle. Tap dancers strategically
relax at the ankle, while classical Indian dancers flex at the ankle. Ballroom
dancers are judged by the clarity of the patterns their feet make on the floor.

Postmodern choreographers mix the ordinary footwork of walking with extraordinary phrases of dancerly movement. In Memphis jookin’, dancers
mix modes as well, even twirling on the tips of their sneakers. How dancers
use their feet to move reveals both their own personal history and the larger
cultural history of the dance form.
But we are not concerned solely with how a dancer gets from here to
there horizontally—that, we know, is assisted by Newton’s 3rd Law of Motion. We are heading in this chapter toward another dimension of dance:
leaving the ground; not only by jumping directly upward, but also by leaping up and out. What resources do dancers use to soar?

Taking Off and Landing
Jumping in dance usually involves three components: the feet, the knees,
and the arms. The feet and the knees, in plié, provide the engine for flight.
Dancers push off through their feet. And just as a spring coils and releases,
the plié that precedes a jump dictates its height. The plié also cushions the
descent. The arms are important as well: note the strategic use of a long
jumper’s arms in flight. Dancers similarly swing their arms so as to add
greater height and distance to a jump.
We have been thinking about feet, so we will focus there. Musicality,
timing, and ability to react all lie in a dancer’s use of the feet. Dancers use
their feet to jump in a variety of ways, depending on the dance form. In
one school of thought, dancers ”roll through” their feet on the way up from
the plié and into the air, and then reverse the sequence on the way down.
”Rolling through” the foot means that once the legs are nearly straight out
of the plié, first the heel leaves the floor, then the ball of the foot presses
through, and then the toes give the final push upward. On the way back
down, the dancer reverses this process by way of contact with the floor: the
toes meet the floor, then the balls of the feet, then the heels.
Perhaps counterintuitively, jumping technique often focuses on the landing, more than on achieving great heights. Rolling through the feet gives
a dancer tremendous control, especially in the descent to the earth. Imagine a cat leaping off a counter to the ground: it does not crash down awkwardly and struggle to recover. Instead, it heads straight into its next action.
Dancers practice rolling through their feet over and over again in order to
build the strength required to land as gracefully as a cat.
Other schools of thought might include landing on the balls of the feet,
or on the full flat foot. Sometimes an ascent has less to do with achieving great height than with coming right back down to execute grounded
rhythms that require deliberate weightiness into the floor. The landing is
frequently more important than the jump itself—adding yet another dimension to the dancer’s duet with gravity.
Think for a moment about how many different ways a dancer with the
full use of both feet has to take off and land:


Taking off from two feet / Landing on two feet
Taking off from two feet / Landing on the left foot
Taking off from two feet / Landing on the right foot
Taking off from the left foot / Landing on two feet
Taking off from the left foot / Landing on the left foot
Taking off from the left foot / Landing on the right foot
Taking off from the right foot / Landing on two feet
Taking off from the right foot / Landing on the left foot
Taking off from the right foot / Landing on the right foot
That’s nine ways to take off and land using the feet. The dancer’s anatomical
structure is to a choreographer what metrical beats are to a poet. When writing sonnets, poets make art out of a set number of beats per line. Similarly,
a choreographer makes art out of limits imposed by the human form. And
we have not yet even covered other ways of taking off and landing, which
might involve the hands, back, shoulders, and even head. Choreographers
can multiply the potential of the human body in ways that many people
have never imagined.
To understand the basic mechanics of landing, try this exercise. Find parallel position. Working gently, perform four jumps straight in a row, practicing rolling through the feet as described above. Setting a metronome will
help keep you on a beat. Jumping a centimeter or two off the floor works
just fine—no need to hit the ceiling. For dancers of different physical abilities, this exercise can also be practiced with the hands against the thighs or a
wall, to help them investigate the effects of pressing through to take off and
cushioning the descent.

Projectile Motion
Once you have caught your breath and recovered from your investigation
into jumping, physics has more surprises for you.
One of the most powerful ideas in the analysis of motion is that action in
one axis can be analyzed independently of action along another axis. First
you must situate the motion—such as a dancer leaping through the air—
within a coordinate system. You can then analyze the vertical component of
the leap, along the y-axis, separately from the horizontal component, along
the x-axis. The link between motion in one axis and motion in the other is
that time is flowing at the same rate in each.
Let’s first introduce the variables you will be manipulating as you work
with modeling motion in physics. You will need variables that allow you to
keep track of a mover’s position at any given point in time. So the motion
must first be put within the context of a coordinate system. You can keep
track of the position of the center of mass of a dancer, or a foot or hand,
using the coordinates x, y, and z. This assumes that you have declared the

directions in which these axes are aligned and the point at which x = y = z
= 0. At any given moment, you can figure out the position of something by
knowing the value of the three coordinates.
To construct the axes, align the direction of gravity with the y-axis, as is
the convention. The direction toward the center of the earth (down) dictates
the negative y direction. Positive y therefore points directly up toward the
If your studio is a square or rectangle, you might find it simplest to let
the x- and z-axes be aligned with the walls. Or the x-axis could be aligned
with the particular direction that a dancer is moving in the motion that you
are analyzing. Set the x = y = z = 0 point at a location that will make your
calculations simplest. You may find that the floor provides the most convenient y = 0 position. Or it might be simplest to set y = 0 at the initial or
final position of a dancer’s center of mass. It takes practice solving problems
to get better at making these decisions. As long as you define your coordinate system consistently within a problem, your calculations should give
you correct answers.

In addition to predicting or keeping track of the position of a person or
thing, we also want to understand the rate at which the position is changing.
This is the velocity. While a position can be given in units of distance, like
meters, velocity deals with the change of that position as a function of time,
so has units of meters over seconds (m/s). You can use vx , vy , and vz to keep
track of the velocity in the x-, y-, and z-axes, respectively.
Finally, we also care about how the velocity of the object changes as a
function of time. This is the object’s acceleration. Since acceleration represents velocity over time, it is calculated in meters per second per second,
which gives m/s2 . We use ax , ay , and az to keep track of the acceleration in
Let’s start with a bite-sized piece of motion, but limit the analysis in a
way that still allows you to do something interesting—not a trivial task,
given how complicated motion in dance is. A set of equations is available

that can be used in cases where acceleration is constant. If you constrain
yourself to analyzing movements when dancers are in the air, the only relevant force acting on them can be gravity, which provides a constant acceleration in negative y in our coordinate system. Assume that the dancers are
close to the earth’s surface, where the constant acceleration due to gravity is
denoted by the variable g, with a value of 9.8 m/s2 in negative y.

Since the dancer is only under the influence of the constant force of gravity in this simplified picture, he or she has two possibilities for jumping:
vertically directly up and down, with motion constrained in y, or sailing
outward, with a horizontal as well as vertical component to the jump. To
simplify the task, arrange the coordinate system so that the x-axis is aligned
with any horizontal motion that occurs; then you can work with just the x
and y coordinates and not worry about z.
Remember that in the chapter on force we had only the force diagram
and information about the acceleration of the object at one particular instant.
Now bring in the past or the future as appropriate, by including in your
model where the jump came from and where it is going. In other words,
you are making a new model in which you will be able to look at changes in
positions and velocities over time. In any given problem, you will need an
initial set of conditions, labeled by i, and a final set of conditions, labeled by
You must set a start time that defines the initial conditions and an ending
time that defines the final conditions. Choose these two times to correspond
to the instants that you want to analyze. You then have the following variables for the start of the movement:

initial position along the x-axis
initial position along the y-axis
initial velocity in the x direction
initial velocity in the y direction
time between the initial and final shapshots


At the end of the movement, each of the position and velocity variables
is repeated, but with the initial i swapped out with final f :

final position along the x-axis
final position along the y-axis
final velocity in the x direction
final velocity in the y direction

Last but not least, we must recognize the effect of the constant acceleration. That acceleration will be equal to g, which is expressed as a = g = 9.8
m/s2 . This is the acceleration due to the force of gravity acting on a mass
near the surface of the earth, and it only acts along the y-axis. There is no
acceleration along the x-axis.

Equations of Constant Acceleration
The framework that you are building is referred to as projectile motion in
physics. Its equations are used for cannonballs, boxes that you toss in the
air, missiles, and more. If you don’t mind tossing yourself into the air—
jumping—then you can also be considered a projectile. Your center of mass
will follow the equations for projectile motion.
We’ll start by introducing all the relevant equations. Then you can work
with them.
xf = xi + vxi t + ax t2
vxf − vxi
ax =
= vxi
+ 2ax (xf − xi )


In equation 9, we see the relationship between position, velocity, and acceleration. We can read through the equation as we would read a sentence: The
final position along an axis (xf ) can be calculated by first taking into account
where you start (xi ). In order to predict where you will end up after a given
period of time, you also need to take into account how fast you were moving
when the clock was started. We accomplish this with the term (vxi t), which
takes your initial speed and multiplies it by the time you will be traveling.
You would now be done if your velocity never changed. We accommodate
changes in velocity—acceleration—and the duration of motion over which
the acceleration acts with the term 21 ax t2 .
This equation will only work if your acceleration is constant over the
time that you consider t. If the situation were more complicated you would
need a more complicated equation.


Before we turn to a calculation, take another pass through the equation,
this time with a different focus: units. You will see that on the left the variable (xf ) has units that match the distance scale, meters. You should therefore expect every term on the right-hand side of the equation to have units
of meters too, or the equals sign will have been a mistake.
The first term, xi , is in units of meters. The second term, vxi t, consists of
a velocity multiplied by a time. With velocity in units of meters per second
(m/s) and time in units of seconds, you can see that the impact of multiplying velocity by time cancels out the seconds, leaving only meters. So far
so good! The third term, 12 ax t2 , begins with a unitless number ( 12 ) that you
can ignore for the purposes of checking unit consistency. What remains is
an acceleration (m/s2 ) multiplied by time squared (s2 ), and, again, here the
seconds cancel out, leaving only meters.
Equation 10 tells us that the acceleration is equal to the difference between the final and initial velocities, divided by the time. A quick glance at
units here shows you that to achieve the units of acceleration (m/s2 ) on the
right side of the equation, take the units of velocity (m/s) and include an
additional time term (in seconds) in the denominator.
Equation 11 allows you to compare the final velocity with the initial velocity. The two will be equal only if the acceleration is 0. Otherwise the initial
velocity squared must be adjusted by a factor of 2a times the total distance
over which the acceleration is applied (xf - xi ). We leave it as an exercise to
you, the reader, to check that we have consistent units on the left and right
sides of the equals sign.
You may be wondering why we need so many equations when each
seems to be relating similar variables in similar contexts. One reason is for
convenience. The first equation can be used regardless of whether you know,
or care about, the final velocity vf . The second equation, on the other hand,
does not rely on the positions or total distance traveled. And the final equation is written in a way that does not use time. When doing a problem with
projectile motion, it is important to first understand which variables you
know and which one(s) you must calculate, so that you can choose the most
appropriate equation to work with.

Jumping Up and Out
If we substitute dancers for objects in our physics analysis, we need to take
into account the techniques with which dancers jump through space. More
specifically, we need to look at how dancers jump not only up, but up and
At this point, you have acquired an arsenal of physics knowledge that
you can translate directly into dance technique. Newton’s 3rd Law of Motion
tells us that the direction of the jumper’s force into the ground directly affects
the direction and angle of the jump. Push straight down into the floor, and
the floor pushes you up. Push at an angle, and the floor pushes you outward
at that angle and you will travel.

Dancers can modulate the same jump to different effects, by changing the
angle of their force into the floor. More loft in the air requires a sharper angle
downward. Covering more distance with any single jump requires pressing
into the floor at a shallower angle. The same principles regarding the use of
the feet and the plié apply.
Swinging the arms down and then up can add an extra element of force.
But the reason why this is effective is counterintuitive. The arms do not
augment the momentum—an image that many dance teachers use to teach
jumps—so much as generate extra force. Swinging your arms upward in
fact gives you greater force into the floor.
You can check this out by swinging your arms forward and back while
standing on a scale. The scale will read above or below your weight at different points in the motion, reflecting the higher and lower degrees of force into
the floor, depending on the acceleration of your arms at any given moment.
Paying attention to the increased pressure between your feet and the floor
as you swing your arms is another way to research this phenomenon. Interestingly, maintaining a constant velocity with the arms does not result in the
same force pressing into the floor. It is only when the arms are accelerating
that the force downward is amplified.
To test all of these ideas, try skipping—a favorite action of adventurous
children and choreographers. Skipping is walking and jumping on a rhythm:
”step, JUMP, step, JUMP, step, JUMP” like so. The jump can either take you
directly upward or help you travel a longer distance.
Skipping is an ideal action with which t