Question

when is the mechanical advantage of a machine equal to its velocity ratio

Answer 1

As velocity ratio or ideal mechanical advantage is a simple ratio of two distances, it also does not have the unit. ... For ideal or perfect machine, work output is equal to the work input.

Answer 2

Mechanical advantage means advantage given by the machine in terms of efforts applied and load lifted by it.
Velocity ratio is the ratio of the velocity at which effort is applied on a machine to the velocity at which load moves.
If D is displacement caused by an effort in time 't' such that 'd' is the displacement caused by the load in the same time, then:
Velocity of effort = D/t
Velocity of load =d/t
Velocity ratio (VR) = velocity of effort /velocity of load
VR=D/t by d/t
VR= D/d