As soon as a traffic light turns green, a car speeds up from rest to 46.0 mi/h with constant acceleration 8.50 mi/h-s. In the adjoining bike lane, a cyclist speeds up from rest to 24.0 mi/h with constant acceleration 12.00 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?
To find the time interval during which the bicycle is ahead of the car, we need to determine when the distance covered by the bicycle is greater than the distance covered by the car.
Let's use the equations of motion to calculate the distances covered by the bicycle and the car.
For the bicycle:
We know that the initial velocity (u) is 0 mi/h, the final velocity (v) is 24.0 mi/h, and the acceleration (a) is 12.00 mi/h-s. We need to find the distance covered (s).
We can use the following equation of motion:
v^2 = u^2 + 2as
Plugging in the known values:
24.0^2 = 0^2 + 2 * 12.00 * s
576 = 24s
s = 576/24
s = 24 mi
So, the distance covered by the bicycle is 24 miles.
For the car:
We know that the initial velocity (u) is 0 mi/h, the final velocity (v) is 46.0 mi/h, and the acceleration (a) is 8.50 mi/h-s. We need to find the distance covered (s).
Using the same equation of motion:
46.0^2 = 0^2 + 2 * 8.50 * s
2116 = 17s
s = 2116/17
s ≈ 124.47 mi
So, the distance covered by the car is approximately 124.47 miles.
Now, we can compare the distances covered by the bicycle and the car.
The bicycle covers 24 miles, and the car covers approximately 124.47 miles.
Since the car covers a greater distance, there is no time interval during which the bicycle is ahead of the car.