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(C) Given that power $P$ is constant.Since $P = F \cdot v = m \cdot a \cdot v = m \cdot \frac{dv}{dt} \cdot v$,we have $m \cdot v \cdot \frac{dv}{dt}

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$t^{3/2}$

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(C) Given that power $P$ is constant.Since $P = F \cdot v = m \cdot a \cdot v = m \cdot \frac{dv}{dt} \cdot v$,we have $m \cdot v \cdot \frac{dv}{dt} = P$.Integrating both sides with respect to time: $\int m \cdot v \cdot dv = \int P \cdot dt$.This gives $\frac{1}{2} m v^2 = P \cdot t$,which implies $v^2 \propto t$,or $v \propto t^{1/2}$.Since velocity $v = \frac{ds}{dt}$,we have $\frac{ds}{dt} \propto t^{1/2}$.Integrating with respect to time: $s = \int t^{1/2} dt = \frac{t^{3/2}}{3/2} = \frac{2}{3} t^{3/2}$.Therefore,displacement $s \propto t^{3/2}$.

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

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