Distance moved as a function of time

[Amith2006]

Sir,
A body is moved along a straight line by a machine delivering constant power. It is said that the distance moved by the particle in time t is proportional to t^(3/2). Can you please explain how this is derived?
Here the symbol ^ represents power.

 

[Meir Achuz]

P=Fv=v m dv/dt=(m/2)d/dt(v^2), so
v^2=(2P/m)t.
Then, dx/dt=(2P/m)^1/2 t^1/2,
and x=(2/3)(2P/m)t^3/2.
This assumes x=-0 and v=0 agt t=0.