How can write a proper equation for this question?

Question

An elevator of mass M is accelerating downward with constant acceleration A. Friction force acting on the elevator is constant and given by f (The initial speed of the elevator is zero.). Find the power generated by the engine of the elevator (in terms of M, A, g, f, and time t).

Answer 1

I am assuming that the downward movement is due to gravity alone, and

that the engine is working to slow down the fall.

The net downward force is then

Mg - f - F, where

Mg is the downward force of gravity,

f is the upward force of friction,

F is the (unknown) upward force of the engine.

This net downward force causes downward acceleration A, therefore

by Newton's Second Law

Mg - f - F = MA

From that we get

F = Mg - MA - f

We want to calculate the power of the force F,

which is the derivative of the work performed by F.

Let

s(t) be the distance (at time t) of the elevator from its initial position.

Then the work, W, performed by F is

W(t) = Fs(t) = (Mg - MA - f)s(t)

The power, P, is the derivative of work:

P(t) = (Mg - MA - f)s'(t)

We can write

s'(t) = At

because the derivative, s'(t), of distance is velocity, and

the velocity of the elevator after time t is At.

So the power generated by the engine at time t is

(Mg - MA - f)At

It is interesting to note under what conditions the power would be 0.

- if A = 0, i.e., the elevator is not moving,

- if A = g - f/M, i.e., if the reduction in the fall we completely due to friction.