(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

A schoolbus travels around a circular path with acceleration a(t)=0.5t m/s/s with t in seconds.

At some point it has a velocity of 8 m/s.

What are the magnitudes of its velocity and acceleration when it has travelled a fourth of the circular track from the point at which it had v = 8 m/s?

Radius of the track is 250m

2. Relevant equations

dv=adt

ds=vdt

3. The attempt at a solution

Using the relation dv=adt,

dv=0.5tdt

I integrate both sides, but get confused as to where I should integrate from since the time the bus has a velocity of 8 m/s is not provided. After consideration, I choose:

v initial is 0, v final is 8 and t initial is 0, t final is just t

After integrating, I solve for t to find

t=6.32 at v=10 and v(t)=0.25t^2

then if a fourth of the circle is travelled, the distance travelled is a fourth of the circumference:

(2*pi*250)/4=125*pi

then ds=vdt=(0.03t^2)dt

We integrate again to find distance travelled as a function of t

integrate ds from 0 to 125*pi and vdt from 6.32 to t, yielding

125*pi=0.083(t)^3-0.083(6.32)^3

From this equation, t=17.08 seconds or it will take 17 seconds to travel a fourth of the way around from this point. From here, we can plug into the a(t) and v(t) equations to answer the question.

I think this is right; however, the first time I solved the problem, I integrated the dv=adt expression from 10 to v on the dv side and 0 to t on the adt side to get v(t)=0.25*t^2+10

I then integrated it again and found the distance as a function of time, solving the time it took to travel from the point at 10 m/s to a point a fourth of the way around the track. What is wrong with this approach?

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Linear Acceleration in a Circular Path

**Physics Forums | Science Articles, Homework Help, Discussion**