A body is moving unidirectionally under the influence of a constant power source. Its displacement in time $$t$$ is proportional to :

We need to find how displacement depends on time for a body under constant power. The constant power relation is $$P = Fv = mav = \text{constant}$$ so $$ma \cdot v = P \Rightarrow m\frac{dv}{dt} \cdot v = P$$ which gives $$mv \, dv = P \, dt$$. Integrating: $$\frac{mv^2}{2} = Pt$$ (starting from rest) and thus $$v = \sqrt{\frac{2Pt}{m}}$$.

Using $$v = \frac{ds}{dt} = \sqrt{\frac{2P}{m}} \cdot t^{1/2}$$ we find $$s = \sqrt{\frac{2P}{m}} \int_0^t t^{1/2} dt = \sqrt{\frac{2P}{m}} \cdot \frac{2}{3}t^{3/2}$$, so $$s \propto t^{3/2}$$.

Displacement is proportional to $$t^{3/2}$$, which matches Option B. Therefore, the answer is Option B.