- #1
MasterTinker
- 13
- 0
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.
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.
