Step 1: State the problem.
A body moves in a straight line while the power supplied to it stays constant. We must find how the distance covered grows with time.
Step 2: Write power in a useful form.
Power is force times velocity, $P = F v$. Force is mass times acceleration, and acceleration is $\dfrac{dv}{dt}$. So $P = m v \dfrac{dv}{dt}$.
Step 3: Separate the variables.
Rearrange to $m\, v\, dv = P\, dt$. This groups velocity terms on one side and time on the other.
Step 4: Integrate both sides.
Integrating gives $\tfrac{1}{2} m v^{2} = P t$. So $v^{2}$ grows in proportion to $t$, which means $v \propto \sqrt{t}$.
Step 5: Link velocity to distance.
Since velocity is $\dfrac{dx}{dt}$, we have $\dfrac{dx}{dt} \propto t^{1/2}$, so $dx \propto t^{1/2}\, dt$.
Step 6: Integrate to get distance.
Integrating $t^{1/2}$ gives $t^{3/2}$. Hence the distance is proportional to $t^{3/2}$. \[ \boxed{t^{3/2}} \]