# Linear Acceleration in a Circular Path

1. Oct 28, 2007

### pecosbill

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

ds=vdt

3. The attempt at a solution

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?

2. Oct 28, 2007

### Staff: Mentor

Doesn't v = 8 m/s?

3. Oct 28, 2007

### Siracuse

Is it just me or there are constants of integration missing?

4. Oct 28, 2007

### pecosbill

well if it starts with a(0)=0 and v(0)=0, there would be no constants.

5. Oct 28, 2007

### pecosbill

also, yes, that 10 is supposed to be an 8.

i'm just trying to establish two things:
a. is it correct to model this as if it were travelling down a straight track with this acceleration?

6. Oct 28, 2007

### pecosbill

and btw, thanks for the help guys. i really appreciate it. this forum is really cool and i plan on participating by responding to other people's questions

7. Oct 28, 2007

### Siracuse

I guess with the info given, you can assume that. Silly me. :X

Don't think there is anything that points this problem to anything else (like, circular motion with centripetal acceleration), so it's okay to model the solution as if the bus were moving in one dimension.

8. Oct 28, 2007

### Staff: Mentor

Yes. The given acceleration of .5t m/s^2 is presumably the tangential acceleration.
How can it be if you used the wrong speed?

9. Oct 28, 2007

### pecosbill

hahahah. okay. thank you.