How Long Does a Bicycle Lead a Car After a Traffic Light Turns Green?

[jcharles513]

Homework Statement

As soon as a traffic light turns green, a car speeds up from rest to 51.0 mi/h with constant acceleration 8.00 mi/h-s. In the adjoining bike lane, a cyclist speeds up from rest to 29.0 mi/h with constant acceleration 12.50 mi/h-s. Each vehicle maintains constant velocity after reaching its cruising speed.
(a) For what time interval is the bicycle ahead of the car?
(b) By what maximum distance does the bicycle lead the car?

Homework Equations

Constant acceleration equations specifically xf=xi+vxi*t+1/2*ax*t^2

The Attempt at a Solution

13.08 seconds as the answer to part A. by finding at what time the velocities are maxed out which is 2.32 and 6.75, respectively the bike and car. After this I set the xf of each equal to each other and solve for t. Any help?

 

[Carid]

A speed/time graph is always a useful tool.

The maximum distance will be when the car reaches the same speed as the bicycle's maximum speed, that is when the bicycle is no longer pulling away from the car.

 

[thomate1]

Your attempt at a solution is correct. To find the time interval in which the bicycle is ahead of the car, you need to find the time at which their positions are equal. This occurs when the final position of the bicycle is equal to the final position of the car, since they both start from rest and have constant velocities after reaching their cruising speeds.

To find this time, you can use the equation xf = xi + vxi*t + 1/2*ax*t^2 for both the bicycle and the car, and set their final positions equal to each other. This will give you a quadratic equation in terms of t, which you can solve to find the time at which their positions are equal.

Once you have this time, you can use it to find the distance that the bicycle is ahead of the car by plugging it into the equation xf = xi + vxi*t + 1/2*ax*t^2 for the bicycle. This will give you the maximum distance that the bicycle leads the car.

In summary, your solution is correct and you just need to use the correct equation to find the time at which their positions are equal and then use that time to find the maximum distance that the bicycle leads the car.