A body is moved along a straight line by a machine delivering constant power. The distance moved by the body in time t is proportional to : 

\(t^{3/2}\)

\(P = F.v = ma.v\)

  = \(m \frac{dv}{dt}.v\)

\(\frac{P}{m}dt = v.dv\)

\(\Rightarrow \int_0^t \frac{P}{m}.dt = \int_0^v v.dv\)

\(\Rightarrow \frac{P}{m}.t = \frac{v^2}{2}\)

\(\Rightarrow v = [\frac{2P}{m}]^{1/2}.t^{1/2}\)

Also, \(s = \int v .dt = \int [\frac{2P}{m}]^{1/2}.t^{1/2}\)

\(s = [\frac{2P}{m}]^{1/2}[\frac{2t^{3/2}}{3}]\)

\(\Rightarrow s \propto t^{3/2}\)