What Factors Influence Acceleration in Non-Uniform Circular Motion?

[MasterTinker]

I know this problem looks easy but my physics mind is out of shape.

Problem

Imagine that you have a car traveling along a circular, horizontal track or radius $r$, with tangential acceleration $a_{lin}$. 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 $$\displaystyle\lim_{r\rightarrow\infty}$$ of the answer in 2?

Attempt at a Solution

1.

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

$$2\pi r=\frac{1}{2}a_{lin}t^2$$

$$t=2\sqrt{\frac{\pi r}{a_{lin}}}$$

$v_{lap}$ is just acceleration by time, therefore:

$$v_{lap}=a_{lin}t$$

$$v_{lap}=a_{lin}\left(2\sqrt{\frac{\pi r}{a_{lin}}}\right)$$

$$v_{lap}=2\sqrt{\pi a_{lin}r}$$

2.

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

$$a_{car}=-a_{centripetal}\hat{r}+a_{tangential}\hat{v}$$

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

$$a_{car}=-\frac{v_{lap}^2}{r}\hat{r}+a_{lin}\hat{v}$$

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

$$a_{car}=-4\pi a_{lin}\hat{r}+a_{lin}\hat{v}$$

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 $r$ approaches $\infty$ 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.

 

[MasterTinker]

By the way I apologize for not placing this in the homework forum. If it's crowding then please redirect it there : )