Average power of car enginer?

Question

A car with mass 1.36 x 10^3 kg starts at rest and accelerates to 10.5 m/s in 13.5 seconds. The force of resistance remains constant at 410 N. What is the average power developed by the cars engine? In units of W.

Answer 1

To solve this problem first we must find the average acceleration of the car.

Average acceleration equals the change in velocity of the car divided by the time it takes for this change to occur.
a = Δv / Δt

We are told that the car accelerates from rest (0 m/s) to a speed of 10.5 m/s in a time of 13.5 second.
Δv = 10.5 m/s
Δt = 13.5 seconds
Therefore, the average acceleration of the car during this time is:
a = (10.5 m/s) / (13.5 s)
a = .778 m/s^2

How far did the car travel during this time period while it was accelerating?
We can use one of the kinematic formulas to find the distance covered by the car while it underwent a constant average acceleration, a, for a time period, t.
Since the car was initially at rest,
d = ½ at^2
We know a = .778 m/s^2 and t = 13.5 secons.
d = ½ (.778 m/s^2) * (13.5 s)^2
d = 70.9 meters

The next step is to find the net force which is acting on the car during this time. To do this we can use Newton’s second law,
Force = mass * acceleration,
F = ma

We know the mass of the car is 1.36 E3 kg and jus just found the average acceleration of the car.
m = 1360 kg
a = .778 m/s^2
So we can find the average force acting on the car over the time it is accelerating as,
F = (1360 kg) * (.778 m/s^2)
F = 1058 Newtons

But this value is the NET force acting on the car. We are told that there is a constant resistive force of 410 Newtons (acting on the opposite direction as the net acceleration). In order for the car to accelerate as we have previously calculated, the engine must be supplying some other, larger, force in the direction of acceleration to counter act this resistive force.
Net force is the sum of all forces acting on the car. In this case, we can assume there are only two force acting on the car, the force supplied by the engine and the resistive force.
We know that the net force should be and we are told that the resistive force is a constant 410 Newtons.
So we can solve for the force supplied by the engine as,
F_engine = F_net + F_resistive
F_engine = (1058 N) + (410 N)
F_engine = 1468 Newtons

Now we need to calculate just how much work the engine did on the car over the distance it applied the force.
A force applied over a distance does work.
Work = Force * distance
We have previously calculated the distance to be 70.9 meters and we have just found that the force supplied by the engine is 1486 Newtons.
So the work preformed by the engine is,
Work = (1468 N) * (70.9 m)
Work = 104081 Joules
Which is the total energy spent by the engine during the period the car was accelerating.

Now, finally, we are ready to calculate the engine’s average power and it will become obvious why we had to go through all the preceding steps.
The engine’s power is the rate of energy output per unit time.
Power = Energy / time
The total work done by the engine is equal to the total energy output by the engine over the time the car was accelerating. We found the total work to be 104081 Joules, and this work was done over a period of 13.5 seconds.
Power = (104081 Joules) / (13.5 seconds)
Power = 7710 Joules per second

Power = 7710 watts

Answer 2

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