As soon as a traffic light turns green, a car speeds up from rest to 57.0 mi/h with constant acceleration 8.00 mi/h/s. In the adjoining bike lane, a cyclist speeds up from rest to 21.0 mi/h with constant acceleration 12.50 mi/h/s. Each vehicle maintains constant velocity after reaching its cruising speed.
To solve this problem, we need to use the equations of motion to determine the time it takes for each vehicle to reach their cruising speed.
Let's start with the car. We know the initial velocity (rest) is 0 mi/h, the final velocity is 57.0 mi/h, and the acceleration is 8.00 mi/h/s. We need to find the time it takes to reach the final velocity.
Using the equation of motion:
v = u + at
Where:
v = final velocity
u = initial velocity
a = acceleration
t = time
We can rearrange the equation to solve for time:
t = (v - u) / a
Substituting the given values:
t = (57.0 mi/h - 0 mi/h) / 8.00 mi/h/s
t = 7.125 seconds (rounded to three decimal places)
Now let's solve for the cyclist. We know the initial velocity (rest) is 0 mi/h, the final velocity is 21.0 mi/h, and the acceleration is 12.50 mi/h/s.
Using the same equation of motion:
t = (v - u) / a
Substituting the given values:
t = (21.0 mi/h - 0 mi/h) / 12.50 mi/h/s
t = 1.68 seconds (rounded to two decimal places)
Therefore, it takes the car approximately 7.125 seconds to reach 57.0 mi/h, and it takes the cyclist approximately 1.68 seconds to reach 21.0 mi/h.