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Motion in One Dimension

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1 Motion in One Dimension
Chapter 2 Motion in One Dimension

2 Kinematics Describes motion while ignoring the agents that caused the motion For now, will consider motion in one dimension Along a straight line Will use the particle model A particle is a point-like object, has mass but infinitesimal size

3 Position The object’s position is its location with respect to a chosen reference point Consider the point to be the origin of a coordinate system In the diagram, allow the road sign to be the reference point

4 Position-Time Graph The position-time graph shows the motion of the particle (car) The smooth curve is a guess as to what happened between the data points

5 Motion of Car Note the relationship between the position of the car and the points on the graph Compare the different representations of the motion

6 Data Table The table gives the actual data collected during the motion of the object (car) Positive is defined as being to the right

7 Alternative Representations
Using alternative representations is often an excellent strategy for understanding a problem For example, the car problem used multiple representations Pictorial representation Graphical representation Tabular representation Goal is often a mathematical representation

8 Displacement Defined as the change in position during some time interval Represented as x x ≡ xf - xi SI units are meters (m) x can be positive or negative Different than distance – the length of a path followed by a particle

9 Distance vs. Displacement – An Example
Assume a player moves from one end of the court to the other and back Distance is twice the length of the court Distance is always positive Displacement is zero Δx = xf – xi = 0 since xf = xi

10 Vectors and Scalars Vector quantities need both magnitude (size or numerical value) and direction to completely describe them Will use + and – signs to indicate vector directions Scalar quantities are completely described by magnitude only

11 Average Velocity The average velocity is rate at which the displacement occurs The x indicates motion along the x-axis The dimensions are length / time [L/T] The SI units are m/s Is also the slope of the line in the position – time graph

12 Average Speed Speed is a scalar quantity
same units as velocity total distance / total time: The speed has no direction and is always expressed as a positive number Neither average velocity nor average speed gives details about the trip described

13 Instantaneous Velocity
The limit of the average velocity as the time interval becomes infinitesimally short, or as the time interval approaches zero The instantaneous velocity indicates what is happening at every point of time

14 Instantaneous Velocity, graph
The instantaneous velocity is the slope of the line tangent to the x vs. t curve This would be the green line The light blue lines show that as t gets smaller, they approach the green line

15 Instantaneous Velocity, equations
The general equation for instantaneous velocity is The instantaneous velocity can be positive, negative, or zero

16 Instantaneous Speed The instantaneous speed is the magnitude of the instantaneous velocity The instantaneous speed has no direction associated with it

17 Vocabulary Note “Velocity” and “speed” will indicate instantaneous values Average will be used when the average velocity or average speed is indicated

18 Analysis Models Analysis models are an important technique in the solution to problems An analysis model is a previously solved problem It describes The behavior of some physical entity The interaction between the entity and the environment Try to identify the fundamental details of the problem and attempt to recognize which of the types of problems you have already solved could be used as a model for the new problem

19 Analysis Models, cont Based on four simplification models
Particle model System model Rigid object Wave

20 Particle Under Constant Velocity
Constant velocity indicates the instantaneous velocity at any instant during a time interval is the same as the average velocity during that time interval vx = vx, avg The mathematical representation of this situation is the equation Common practice is to let ti = 0 and the equation becomes: xf = xi + vx t (for constant vx)

21 Particle Under Constant Velocity, Graph
The graph represents the motion of a particle under constant velocity The slope of the graph is the value of the constant velocity The y-intercept is xi

22 Average Acceleration Acceleration is the rate of change of the velocity Dimensions are L/T2 SI units are m/s² In one dimension, positive and negative can be used to indicate direction

23 Instantaneous Acceleration
The instantaneous acceleration is the limit of the average acceleration as t approaches 0 The term acceleration will mean instantaneous acceleration If average acceleration is wanted, the word average will be included

24 Instantaneous Acceleration -- graph
The slope of the velocity-time graph is the acceleration The green line represents the instantaneous acceleration The blue line is the average acceleration

25 Graphical Comparison Given the displacement-time graph (a)
The velocity-time graph is found by measuring the slope of the position-time graph at every instant The acceleration-time graph is found by measuring the slope of the velocity-time graph at every instant

26 Acceleration and Velocity, 1
When an object’s velocity and acceleration are in the same direction, the object is speeding up When an object’s velocity and acceleration are in the opposite direction, the object is slowing down

27 Acceleration and Velocity, 2
Images are equally spaced. The car is moving with constant positive velocity (shown by red arrows maintaining the same size) Acceleration equals zero

28 Acceleration and Velocity, 3
Images become farther apart as time increases Velocity and acceleration are in the same direction Acceleration is uniform (violet arrows maintain the same length) Velocity is increasing (red arrows are getting longer) This shows positive acceleration and positive velocity

29 Acceleration and Velocity, 4
Images become closer together as time increases Acceleration and velocity are in opposite directions Acceleration is uniform (violet arrows maintain the same length) Velocity is decreasing (red arrows are getting shorter) Positive velocity and negative acceleration

30 Acceleration and Velocity, final
In all the previous cases, the acceleration was constant Shown by the violet arrows all maintaining the same length The diagrams represent motion of a particle under constant acceleration A particle under constant acceleration is another useful analysis model

31 Graphical Representations of Motion
Observe the graphs of the car under various conditions Note the relationships among the graphs Set various initial velocities, positions and accelerations

32 Kinematic Equations – summary

33 Kinematic Equations The kinematic equations can be used with any particle under uniform acceleration. The kinematic equations may be used to solve any problem involving one-dimensional motion with a constant acceleration You may need to use two of the equations to solve one problem Many times there is more than one way to solve a problem

34 Kinematic Equations, specific
For constant a, Can determine an object’s velocity at any time t when we know its initial velocity and its acceleration Assumes ti = 0 and tf = t Does not give any information about displacement

35 Kinematic Equations, specific
For constant acceleration, The average velocity can be expressed as the arithmetic mean of the initial and final velocities

36 Kinematic Equations, specific
For constant acceleration, This gives you the position of the particle in terms of time and velocities Doesn’t give you the acceleration

37 Kinematic Equations, specific
For constant acceleration, Gives final position in terms of velocity and acceleration Doesn’t tell you about final velocity

38 Kinematic Equations, specific
For constant a, Gives final velocity in terms of acceleration and displacement Does not give any information about the time

39 When a = 0 When the acceleration is zero,
vxf = vxi = vx xf = xi + vx t The constant acceleration model reduces to the constant velocity model

40 Graphical Look at Motion: displacement – time curve
The slope of the curve is the velocity The curved line indicates the velocity is changing Therefore, there is an acceleration

41 Graphical Look at Motion: velocity – time curve
The slope gives the acceleration The straight line indicates a constant acceleration

42 Graphical Look at Motion: acceleration – time curve
The zero slope indicates a constant acceleration

43 Graphical Motion with Constant Acceleration
A change in the acceleration affects the velocity and position Note especially the graphs when a = 0

44 Test Graphical Interpretations
Match a given velocity graph with the corresponding acceleration graph Match a given acceleration graph with the corresponding velocity graph(s)

45 Galileo Galilei 1564 – 1642 Italian physicist and astronomer
Formulated laws of motion for objects in free fall Supported heliocentric universe

46 Freely Falling Objects
A freely falling object is any object moving freely under the influence of gravity alone. It does not depend upon the initial motion of the object Dropped – released from rest Thrown downward Thrown upward

47 Acceleration of Freely Falling Object
The acceleration of an object in free fall is directed downward, regardless of the initial motion The magnitude of free fall acceleration is g = 9.80 m/s2 g decreases with increasing altitude g varies with latitude 9.80 m/s2 is the average at the Earth’s surface The italicized g will be used for the acceleration due to gravity Not to be confused with g for grams

48 Acceleration of Free Fall, cont.
We will neglect air resistance Free fall motion is constantly accelerated motion in one dimension Let upward be positive Use the kinematic equations with ay = -g = -9.80 m/s2

49 Free Fall – an object dropped
Initial velocity is zero Let up be positive Use the kinematic equations Generally use y instead of x since vertical Acceleration is ay = -g = m/s2 vo= 0 a = -g

50 Free Fall – an object thrown downward
ay = -g = m/s2 Initial velocity  0 With upward being positive, initial velocity will be negative vo≠ 0 a = -g

51 Free Fall -- object thrown upward
Initial velocity is upward, so positive The instantaneous velocity at the maximum height is zero ay = -g = m/s2 everywhere in the motion v = 0 vo≠ 0 a = -g

52 Thrown upward, cont. The motion may be symmetrical
Then tup = tdown Then v = -vo The motion may not be symmetrical Break the motion into various parts Generally up and down

53 Free Fall Example Initial velocity at A is upward (+) and acceleration is -g (-9.8 m/s2) At B, the velocity is 0 and the acceleration is -g (-9.8 m/s2) At C, the velocity has the same magnitude as at A, but is in the opposite direction The displacement is –50.0 m (it ends up 50.0 m below its starting point)

54 Kinematic Equations from Calculus
Displacement equals the area under the velocity – time curve The limit of the sum is a definite integral

55 Kinematic Equations – General Calculus Form

56 Kinematic Equations – Calculus Form with Constant Acceleration
The integration form of vf – vi gives The integration form of xf – xi gives

57 General Problem Solving Strategy
Conceptualize Categorize Analyze Finalize

58 Problem Solving – Conceptualize
Think about and understand the situation Make a quick drawing of the situation Gather the numerical information Include algebraic meanings of phrases Focus on the expected result Think about units Think about what a reasonable answer should be

59 Problem Solving – Categorize
Simplify the problem Can you ignore air resistance? Model objects as particles Classify the type of problem Substitution Analysis Try to identify similar problems you have already solved What analysis model would be useful?

60 Problem Solving – Analyze
Select the relevant equation(s) to apply Solve for the unknown variable Substitute appropriate numbers Calculate the results Include units Round the result to the appropriate number of significant figures

61 Problem Solving – Finalize
Check your result Does it have the correct units? Does it agree with your conceptualized ideas? Look at limiting situations to be sure the results are reasonable Compare the result with those of similar problems

62 Problem Solving – Some Final Ideas
When solving complex problems, you may need to identify sub-problems and apply the problem-solving strategy to each sub-part These steps can be a guide for solving problems in this course

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