A 1.50 103-kg car starts from rest and accelerates uniformly to 18.5 m/s in 11.3 s. Assume that air resistance remains constant at 400 N during this time.
(a) Find the average power developed by the engine.
1 W
(b) Find the instantaneous power output of the engine at t = 11.3 s, just before the car stops accelerating.
2 W
To find the average power developed by the engine, we can use the formula:
Average power = Work / Time
First, let's find the work done on the car. The work done on an object is given by the product of the force applied and the distance covered. In this case, the force applied is the net force acting on the car, which is the sum of the engine force and the air resistance force. The distance covered is determined by the acceleration of the car.
The formula for net force is:
Net force = mass × acceleration
Given:
Mass (m) = 1.50 × 10^3 kg
Acceleration (a) = Change in velocity / Time = (Final velocity - Initial velocity) / Time = (18.5 m/s - 0 m/s) / 11.3 s
Now, let's calculate the acceleration:
Acceleration (a) = 18.5 m/s / 11.3 s = 1.637 m/s²
Now, let's calculate the net force:
Net force = mass × acceleration
Net force = 1.50 × 10^3 kg × 1.637 m/s² = 2,455.5 N
Next, let's calculate the work done:
Work = Force × Distance
The distance covered can be calculated using the formula:
Distance = (Initial velocity × Time) + (1/2 × acceleration × Time²)
Distance = (0 m/s × 11.3 s) + (1/2 × 1.637 m/s² × (11.3 s)²)
Now, calculate the distance:
Distance = 1/2 × 1.637 m/s² × (11.3 s)² = 102.1 m
Now, let's calculate the work done:
Work = Force × Distance = (2,455.5 N + 400 N) × 102.1 m
Work = 2,855.5 N × 102.1 m = 291,691.55 J
Finally, let's calculate the average power:
Average power = Work / Time = 291,691.55 J / 11.3 s ≈ 25,821.78 W
Therefore, the average power developed by the engine is approximately 25,821.78 Watts.
To find the instantaneous power output of the engine at t = 11.3 s, just before the car stops accelerating, we need to find the net force and the velocity at that time.
We already calculated the net force as 2,455.5 N. Now, let's calculate the velocity at t = 11.3 s.
Velocity = Initial velocity + (Acceleration × Time)
Velocity = 0 m/s + (1.637 m/s² × 11.3 s) ≈ 18.49 m/s
Now, let's calculate the instantaneous power output:
Instantaneous power = Force × Velocity = (2,455.5 N + 400 N) × 18.49 m/s
Instantaneous power ≈ 2,855.5 N × 18.49 m/s ≈ 52,802.095 W
Therefore, the instantaneous power output of the engine at t = 11.3 s, just before the car stops accelerating, is approximately 52,802.095 Watts.