# Rotational motion question

sheepcountme

## Homework Statement

Two cars race around a circular track. Car A accelerates at 0.340 rad/s2 around the track, and car B at 0.270 rad/s2. They start at the same place on the track and car A lets the slower-to-accelerate car B start first. Car B starts at time t = 0. When car A starts, car B has an angular velocity of 1.40 rad/s. At what time does car A catch up to car B?

## Homework Equations

rotational motion equations

## The Attempt at a Solution

I attempted to use rotational motion equations and set up

theta car A=theta car B
since they will be in the same place (theta) at the same time.

as in: omega initial A x time + 1/2 alpha A time^2 = omega initial B x time + 1/2 alpha A time^2

So: 0(t) + 1/2(.340)t^2=1.4(t)+1/2(.270)t^2
and then I solved for time using quadratic theorem and got 40 seconds but this was incorrect. Could you tell me where I went wrong?? Thanks!

aim1732
Is the angle really equal ? The second car already has traversed some angle when the other starts. So angle by faster car > angle by slower car.

dpeagler
You first have to find delta theta to see how far car B has gone. Then find how long it took to traverse that distance. Then plug your delta theta into the equations for car A with its respective angular velocity then add the two separate times (first time while car A was waiting + time it took car A to catch up) should be way less than 40 seconds if I'm not mistaking.

sheepcountme
Okay, so I used omegafinal^2=omegainitial^2+2(alpha*deltatheta) to find delta theta for car B coming to delta theta=3.63rad

Then omegafinal=omegainitial+alphat to find the time for car B to reach this point, getting t=5.19s

Then I set the previous delta theta equal to an equation for Car A:
(3.63)=omegainitialt+1/2(alpha)t^2
3.63=0+1/2(.34)t^2
and solved for t getting 4.62 seconds

and so I added this 4.62 to 5.19 getting 9.81 however this was incorrect...

dpeagler
The first two parts you did are correct ( when you found the radians and the time ), as far as I can tell.

omega initial A x time + 1/2 alpha A time^2 = omega initial B x time + 1/2 alpha A time^2

You didn't put your Theta initial B as 3.63 radians. This should fix your problem. Sorry I haven't had time to sit down and work it out to make sure this is the mistake, but I'm pretty sure.

sheepcountme
So, using the equation above, I get
1/2(.340)t^2=3.63t+1/2(.270)t^2
0=-.035t^2+3.63t

and time ends up being 103.72s This seems much too big...I hate needing someone to hold my hand through this, but there's really something I'm missing.

dpeagler
You're still not using that equation correctly.

$$\theta$$= $$\theta$$$$_{int}$$ + $$\omega$$$$_{int}$$t + $$\frac{1}{2}$$$$\alpha$$t$$^{2}$$

(sorry about the formatting still trying to get used to LaTeX.)

You then set them equal to each other. You put your radians in the wrong place on the equation you listed. You put them where the initial angular velocity goes. This should bring your answer down by around a factor of two.
The text passage is a discussion thread from a physics homework help forum, where a student is asking for help with a rotational motion problem involving two cars racing on a circular track. The student provides their attempt at solving the problem, and other users provide feedback and suggestions.
single_question_with_answer

1. the passage contains a single mathematical question: "at what time does car a catch up to car b?"
2. an answer to this question is provided in the text, but it is incorrect. the student initially calculates the time to be 40 seconds, which is incorrect. they then try another approach and get 9.81 seconds, which is also incorrect. the other users provide feedback and suggestions, but a correct answer is not explicitly stated in the text.
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# Rotational motion question

sheepcountme

## Homework Statement

Two cars race around a circular track. Car A accelerates at 0.340 rad/s2 around the track, and car B at 0.270 rad/s2. They start at the same place on the track and car A lets the slower-to-accelerate car B start first. Car B starts at time t = 0. When car A starts, car B has an angular velocity of 1.40 rad/s. At what time does car A catch up to car B?

## Homework Equations

rotational motion equations

## The Attempt at a Solution

I attempted to use rotational motion equations and set up

theta car A=theta car B
since they will be in the same place (theta) at the same time.

as in: omega initial A x time + 1/2 alpha A time^2 = omega initial B x time + 1/2 alpha A time^2

So: 0(t) + 1/2(.340)t^2=1.4(t)+1/2(.270)t^2
and then I solved for time using quadratic theorem and got 40 seconds but this was incorrect. Could you tell me where I went wrong?? Thanks!

aim1732
Is the angle really equal ? The second car already has traversed some angle when the other starts. So angle by faster car > angle by slower car.

dpeagler
You first have to find delta theta to see how far car B has gone. Then find how long it took to traverse that distance. Then plug your delta theta into the equations for car A with its respective angular velocity then add the two separate times (first time while car A was waiting + time it took car A to catch up) should be way less than 40 seconds if I'm not mistaking.

sheepcountme
Okay, so I used omegafinal^2=omegainitial^2+2(alpha*deltatheta) to find delta theta for car B coming to delta theta=3.63rad

Then omegafinal=omegainitial+alphat to find the time for car B to reach this point, getting t=5.19s

Then I set the previous delta theta equal to an equation for Car A:
(3.63)=omegainitialt+1/2(alpha)t^2
3.63=0+1/2(.34)t^2
and solved for t getting 4.62 seconds

and so I added this 4.62 to 5.19 getting 9.81 however this was incorrect...

dpeagler
The first two parts you did are correct ( when you found the radians and the time ), as far as I can tell.

omega initial A x time + 1/2 alpha A time^2 = omega initial B x time + 1/2 alpha A time^2

You didn't put your Theta initial B as 3.63 radians. This should fix your problem. Sorry I haven't had time to sit down and work it out to make sure this is the mistake, but I'm pretty sure.

sheepcountme
So, using the equation above, I get
1/2(.340)t^2=3.63t+1/2(.270)t^2
0=-.035t^2+3.63t

and time ends up being 103.72s This seems much too big...I hate needing someone to hold my hand through this, but there's really something I'm missing.

dpeagler
You're still not using that equation correctly.

$$\theta$$= $$\theta$$$$_{int}$$ + $$\omega$$$$_{int}$$t + $$\frac{1}{2}$$$$\alpha$$t$$^{2}$$

(sorry about the formatting still trying to get used to LaTeX.)

You then set them equal to each other. You put your radians in the wrong place on the equation you listed. You put them where the initial angular velocity goes. This should bring your answer down by around a factor of two.
user
The text passage is a discussion thread from a physics homework help forum, where a student is asking for help with a rotational motion problem involving two cars racing on a circular track. The student provides their attempt at solving the problem, and other users provide feedback and suggestions.

TYPE: SINGLE_QUESTION_WITH_ANSWER

1. The passage contains a single mathematical question: "At what time does car A catch up to car B?"
2. An answer to this question is provided in the text, but it is incorrect. The student initially calculates the time to be 40 seconds, which is incorrect. They then try another approach and get 9.81 seconds, which is also incorrect. The other users provide feedback and suggestions, but a correct answer is not explicitly stated in the text.