# Minimum distance around a track

better361

## Homework Statement

A motorbike starts from rest at time t=0 and begins to accelerate around a circular track of radius 10m. Eventually, at time t=t1 the motorbike reaches the maximum velocity possible without slipping off the track. What's the minimum length in meters the motorbike must travel between t=0 and t=t1?

## Homework Equations

mv^2/r=Force of friction in radial direction
m*dv/dt=Force of friction in tangential direction

## The Attempt at a Solution

Here is the solution that was given but I did not understand it.
Call β the the angle formed by friction with respect to the radial axis by taking the arctan of (friction in tangential/friction in radial). So, mv^2/r=F*sinβ. Differentiating w.r.t, the equation is now (2mv/r)*dv/dt=F*cosβ*dβ/dt. The second equation becomes, with the substitution of β ,
m*dv/dt=F*cosβ. Dividing these two equations, it now becomes 2v/r=dβ/dt. v/r is the rate of change of the angle,θ, formed from the traveled distance(the arclength) over the radius. Making this substitution and canceling the differential dt, the equation is now dθ=dβ/2.
Then, as β goes from 0 to pi/2, the distance traveled is s=θ*r=pi/4*r=7.85m.

Is there another way to solve this problem? The part that I am still confused with is how to realize to take the derivative and divide the equations.
The text passage contains a physics problem that involves a motorbike accelerating on a circular track and asks for the minimum distance the motorbike must travel to reach its maximum velocity without slipping. The solution provided uses calculus and trigonometry to derive the answer.
single_question_with_answer
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# Minimum distance around a track

better361

## Homework Statement

A motorbike starts from rest at time t=0 and begins to accelerate around a circular track of radius 10m. Eventually, at time t=t1 the motorbike reaches the maximum velocity possible without slipping off the track. What's the minimum length in meters the motorbike must travel between t=0 and t=t1?

## Homework Equations

mv^2/r=Force of friction in radial direction
m*dv/dt=Force of friction in tangential direction

## The Attempt at a Solution

Here is the solution that was given but I did not understand it.
Call β the the angle formed by friction with respect to the radial axis by taking the arctan of (friction in tangential/friction in radial). So, mv^2/r=F*sinβ. Differentiating w.r.t, the equation is now (2mv/r)*dv/dt=F*cosβ*dβ/dt. The second equation becomes, with the substitution of β ,
m*dv/dt=F*cosβ. Dividing these two equations, it now becomes 2v/r=dβ/dt. v/r is the rate of change of the angle,θ, formed from the traveled distance(the arclength) over the radius. Making this substitution and canceling the differential dt, the equation is now dθ=dβ/2.
Then, as β goes from 0 to pi/2, the distance traveled is s=θ*r=pi/4*r=7.85m.

Is there another way to solve this problem? The part that I am still confused with is how to realize to take the derivative and divide the equations.
user
The text passage contains a physics problem that involves a motorbike accelerating on a circular track and asks for the minimum distance the motorbike must travel to reach its maximum velocity without slipping. The solution provided uses calculus and trigonometry to derive the answer.

TYPE: SINGLE_QUESTION_WITH_ANSWER