A 1.50 ✕ 103-kg car starts from rest and accelerates uniformly to 19.3 m/s in 11.7 s. Assume that air resistance remains constant at 400 N during this time.
(a) Find the average power developed by the engine.
W
(b) Find the instantaneous power output of the engine at t = 11.7 s, just before the car stops accelerating.
(a) To find the average power developed by the engine, we need to use the formula for power: Power = Work / Time.
To find the work done by the engine, we need to calculate the net force acting on the car.
First, let's find the net force. Using Newton's second law (F = ma), we can calculate the acceleration:
a = (vf - vi) / t = (19.3 m/s - 0 m/s) / 11.7 s = 1.65 m/s^2
Now, we can calculate the net force using the formula:
Fnet = ma = (1.50 ✕ 10^3 kg) * (1.65 m/s^2) = 2475 N
Since the car experiences air resistance of 400 N, we need to subtract this from the net force to get the force developed by the engine:
Fengine = Fnet - Fair = 2475 N - 400 N = 2075 N
The work done by the engine is equal to the force multiplied by the displacement:
Work = Fengine * d
However, we need to find the displacement first. We can use the equation of motion:
vf^2 = vi^2 + 2ad
Rearranging the equation, we get:
d = (vf^2 - vi^2) / (2a)
d = (19.3 m/s)^2 / (2 * 1.65 m/s^2) = 115.70 m
Now, we can calculate the work done by the engine:
Work = Fengine * d = (2075 N) * (115.70 m) = 239,982.50 J (joules)
Finally, we can find the average power by dividing the work by the time:
Average Power = Work / Time = 239,982.50 J / 11.7 s ≈ 20,512 W (watts)
Therefore, the average power developed by the engine is approximately 20,512 W.
(b) To find the instantaneous power output of the engine at t = 11.7 s, just before the car stops accelerating, we need to find the force at that instant and multiply it by the instantaneous velocity.
Given that the acceleration remains constant, the velocity at t = 11.7 s is equal to the final velocity:
v = 19.3 m/s
To find the force developed by the engine at t = 11.7 s, we can use Newton's second law:
F = ma = (1.50 ✕ 10^3 kg) * (1.65 m/s^2) = 2475 N
The instantaneous power output of the engine is the product of the force and the velocity:
Instantaneous Power = F * v = (2475 N) * (19.3 m/s) ≈ 47,827.5 W (watts)
Therefore, the instantaneous power output of the engine at t = 11.7 s is approximately 47,827.5 W.