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

\(t^{3/2}\)

Let us assume that the displacement of the body is directly proportional to \(t^n\), i.e. :

\(s = Kt^n\)

Thus, the velocity : \(v = \frac{ds}{dt} = Kn t^{n-1}\)

and the acceleration : \(a = \frac{dv}{dt} = Kn(n-1)t^{n-2}\)

The Force : \(F = ma = mKn(n-1)t^{n-2}\)

Hence, Power : \(P = Fv = (mKn(n-1)t^{n-2})(Kn t^{n-1})\)

\(P = mKn^2(n-1)t^{2n-3}\)

As power is constant, Thus, independent of time.

Hence : 2n - 3 = 0

\(\Rightarrow n = \frac{3}{2}\)

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