Variable mass

Let the drop fall through a height ∂x in time ∂t

m = ∂M∂t
∂M = m∂t
or,M = ∫m∂t =mt.............(1)

∂x∂t= ∂gt²/2∂t=gt

∂x = gt ∂t...........(2)

∂F = g∂M = mg ∂t..............(3)

We know,

W = F*x, since F and x changes with time we have to integrate twice

∂²W = ∂F*∂x = mg ∂t *gt ∂t = mg²t ∂t²

Therefore W = ∫∫mg²t ∂t²
Integrating it once we get,
W = ∫mg²t²2 ∂t = mg²t³6

In the integrals the limits are from 0 to t.

From (1) we get that,
W = M³g²6m² which is the desired result.