Power is the rate at which work is expended,

     P = ΔU/Δt = dU/dt

Recall that work, dU, is simply the force through the distance. If the force is constant, the can be written as

     dU/dt = F · d(r)/dt

or simply,

Notice that the force and direction need to be parallel. The dot product forces the force vector and velocity vector to be parallel.

Mechanical efficiency is the ratio of the "energy out" to the "energy in" of a system. This accounts for energy lost due to system in-efficiencies such as friction.

Efficiency can also be written in terms of power,