Deriving the Maximum Speed Equation for Elevator Motors

[heart_of_fire]

A motor of peak power (p) raises an elevator of mass (m)

a)Show that the maximum speed at which the elevator can be raised is P/mg

I guess my question is how did they get P/mg as as the equation for maximum speed?

 

[RoyalCat]

The power of the motor goes towards raising the elevator.

Power is defined as the rate at which work is done. $$P=\frac{dW}{dt}$$
Consider the fact that work is defined as $$dW=\vec F \cdot \vec dx$$ and try to find an expression relating the power and the velocity. The answer in the book should point towards it quite strongly.