Final answer:
The rate at which the engine is doing work, which is the instantaneous power, is given by P = (ma + R)V where m is the mass, a is the acceleration, V is the velocity, and R is the external resistive force.
Step-by-step explanation:
The question asks us to find the rate at which the engine of a car with mass m, accelerating along a straight level road at acceleration a against a constant external resistive force R with a velocity V, is doing work. The work done by the engine in a time dt is the force exerted by the engine times the distance covered in time dt.
The force exerted by the engine needs to overcome the external resistive force R as well as provide the necessary acceleration for mass m, so it is F = ma + R. The distance covered in time dt is Vdt. Therefore, the work done dW in time dt is this force times the distance, or dW = (ma + R)Vdt. The rate of doing work, which is power P, is P = dW/dt = (ma + R)V.
When a car is driven with acceleration along a straight level road against a constant external resistive force, the rate at which the engine is doing work is given by the product of the force exerted by the engine and the velocity of the car.
The work done by the engine is equal to the power supplied by the engine. The power can be calculated using the formula:
Power (P) = Force (F) x Velocity (V)
Therefore, the rate at which the engine is doing work is equal to the power supplied by the engine.