I know this problem looks easy but my physics mind is out of shape.(adsbygoogle = window.adsbygoogle || []).push({});

Problem

Imagine that you have a car traveling along a circular, horizontal track or radius [itex]r[/itex], with tangential acceleration [itex]a_{lin}[/itex]. If the car begins moving around the track with velocity 0 m/s:

1. What is the tangential speed of the car after one lap?

2. What is the acceleration of the car after it completes one lap?

3. What is the [tex]\displaystyle\lim_{r\rightarrow\infty}[/tex] of the answer in 2?

Attempt at a Solution

1.

In order to calculate the tangential speed of the car after one lap, [itex]v_{lap}[/itex], I first calculate the time it would take for it to complete a lap. The distance around the track is [itex]2\pi r[/itex], and using a kinematics equation:

[tex]2\pi r=\frac{1}{2}a_{lin}t^2[/tex]

[tex]t=2\sqrt{\frac{\pi r}{a_{lin}}}[/tex]

[itex]v_{lap}[/itex] is just acceleration by time, therefore:

[tex]v_{lap}=a_{lin}t[/tex]

[tex]v_{lap}=a_{lin}\left(2\sqrt{\frac{\pi r}{a_{lin}}}\right)[/tex]

[tex]v_{lap}=2\sqrt{\pi a_{lin}r}[/tex]

2.

If I define [itex]\hat{r}[/itex] to be the unit vector pointing directly outwards from the center of the circle, and [itex]\hat{v}[/itex] to be the unit vector pointing in the direction of the car's tangential motion, then the acceleration of the car, [itex]a_{car}[/itex], is:

[tex]a_{car}=-a_{centripetal}\hat{r}+a_{tangential}\hat{v}[/tex]

The centripetal acceleration is proportional to the car's velocity and the radius of the track, while the tangential acceration is just [itex]a_{lin}[/itex], therefore:

[tex]a_{car}=-\frac{v_{lap}^2}{r}\hat{r}+a_{lin}\hat{v}[/tex]

[tex]a_{car}=-\frac{\left(2\sqrt{\pi a_{lin}r}\right)^2}{r}\hat{r}+a_{lin}\hat{v}[/tex]

[tex]a_{car}=-4\pi a_{lin}\hat{r}+a_{lin}\hat{v}[/tex]

3.

Uh-oh

What's Wrong?

I don't see any problem with the way I derived the velocity of the car after one lap. I think how I did the car's acceleration is okay, but what I don't understand is that it is not based on the radius of the track (does this seem weird to anyone else?) and I can't figure out how to interpret the limit required in part 3.

What I would think is that as [itex]r[/itex] approaches [itex]\infty[/itex] the car will have an infinite speed by the time it completes a lap, and therefore have an infinite centripetal acceleration. Or would the car not be able to complete a lap? Or would the centripetal acceleration be 0 as a circle of infinite radius becomes a line? I really don't know what to think, please help me interpret this problem.

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# Homework Help: Non-Uniform Circular Motion

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