A 1.50 ✕ 103 kg car starts from rest and accelerates uniformly to 17.1 m/s in 12.7 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 = 12.7 s, just before the car stops accelerating.
W
To find the average power developed by the engine, we can use the work-energy principle. The work done on the car is equal to the change in kinetic energy. Given the mass of the car (m = 1.50 × 10^3 kg), the initial velocity (v₁ = 0 m/s), the final velocity (v₂ = 17.1 m/s), and the time taken (t = 12.7 s), we can calculate the average power using the following steps:
Step 1: Calculate the change in kinetic energy
The change in kinetic energy (ΔKE) can be calculated as:
ΔKE = 1/2 * m * (v₂^2 - v₁^2)
Substituting the values:
ΔKE = 1/2 * (1.50 × 10^3 kg) * (17.1 m/s)^2
Step 2: Calculate the work done
The work done (W) is equal to the change in kinetic energy:
W = ΔKE
Step 3: Calculate the average power
The average power (P) is equal to the work done divided by the time taken:
P = W / t
Substituting the values:
P = (1/2 * (1.50 × 10^3 kg) * (17.1 m/s)^2) / 12.7 s
Now, let's calculate the average power.
(a) Average Power:
P = (1/2 * (1.50 × 10^3 kg) * (17.1 m/s)^2) / 12.7 s
P ≈ 1776 W
Therefore, the average power developed by the engine is approximately 1776 Watts.
To find the instantaneous power output of the engine at t = 12.7 s, just before the car stops accelerating, we need to account for the constant air resistance (400 N).
Step 4: Calculate the net force
The net force acting on the car is given by:
F_net = m * a
Knowing that the air resistance remains constant at 400 N, the net force can be written as:
F_net = 400 N
Step 5: Calculate the acceleration
The acceleration (a) can be calculated as:
a = (v₂ - v₁) / t
Substituting the values:
a = (17.1 m/s - 0 m/s) / 12.7 s
Step 6: Calculate the applied force
The applied force (F_applied) can be calculated as:
F_applied = F_net - F_resistance
Substituting the values:
F_applied = 400 N - 400 N = 0 N
Step 7: Calculate the instantaneous power
The instantaneous power (P) is equal to the product of the applied force and the velocity:
P = F_applied * v
Substituting the values:
P = 0 N * 17.1 m/s = 0 W
Therefore, the instantaneous power output of the engine at t = 12.7 s, just before the car stops accelerating, is 0 Watts.