Rotational dynamics of two cars

[zoner7]

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 let's 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?

The Attempt at a Solution

So first I calculated the time that car B accelerates before car A begins to move using acceleration of car B and its final angular velocity.

My time value is 5.185

Then I calculated the distance that car B travels until it attains an angular velocity of 1.4.

My distance is 3.629.

Then I used the linear motion equations to set the final distances of cars A and B to one another, resulting in the following equation after simplified:

1/2 Alpha(car A) t^2 - 1/2 Alpha(car B) t^2 - Omega (initial car B) t - Theta (car B).

After plugging in values, I used the quadratic equation and found a time value of 42.44 seconds.

I added the time that Car B travels before it reaches car A: 5.185 + 42.44 and found an ultimate time of 47.625.

But if I plug each the smaller time into car A's linear motion equation and the larger into Car B's, the distances are unequal...

help...?

Thank you in advance.

 

[LowlyPion]

When you plug the longer time into the slower cars equation it's changed. The proper equation for that car then has no initial conditions.

X = 1/2*.27*(47.625)² = 1/*.34*(42.44)²

 

[thomate1]

I would first like to commend you on your attempt to solve this problem by using the appropriate equations and values. Your approach seems logical and well thought out. However, I believe there may be a slight error in your calculations that is resulting in the unequal distances.

When calculating the time that car B accelerates before car A begins to move, you used the final angular velocity of car B to calculate the time. However, since car B is accelerating, its angular velocity would not remain constant until car A starts moving. Therefore, you would need to use the average angular velocity of car B during the time it takes for car A to start moving. This would result in a slightly different time value and ultimately affect the final distance calculation.

Additionally, when using the linear motion equations to set the final distances of cars A and B to be equal, you would also need to consider the initial angular velocity of car B, which is 1.40 rad/s. This would change the equation slightly and result in a different time value.

I would suggest reviewing your calculations and making sure you are using the correct values and equations. It is also helpful to double check your work and make sure all units are consistent throughout.

I hope this helps and wish you the best of luck in solving this problem.