Derive a relationship between mechanical advantage, velocity ratio and efficiency of a machine

Suppose a machine overcomes a load L by the application of effort E.

Let,

the displacement of effort = d_E and

the displacement of the load = d_L in time t.

Work Input = E × d_E

work Output = L × d_L

Efficiency (η) = E × d_E\over L × d_L

⇒ η = {L\over E}×{d_L\over d_E}

⇒ η = {L\over E}×{1\over {d_E\over d_L}}

⇒ η = MA × {1\over VR}

⇒ η = MA \over VR

⇒ MA = η × VR

Hence, the mechanical advantage of a machine is equal to the product of its efficiency and velocity ratio.