A constant power delivering machine has towed a box, which was initially at rest, along a horizontal straight line. The distance moved by the box in time ‘t’ is proportional to:

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$t^{3 / 2}$

$t$

$t^{2 / 3}$

$t^{1 / 2}$

answer is C.

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Detailed Solution

$\begin{array}{ll} P = F v \\ \Rightarrow P = m a v \\ \Rightarrow P = m v \frac{d v}{d x} v \\ \Rightarrow P \int d x = m \int v^{2} \cdot d v \\ \Rightarrow v^{3} \propto x \\ \Rightarrow v \propto x^{\frac{1}{3}} \\ \Rightarrow \frac{d x}{d t} \propto x^{\frac{1}{3}} \\ \Rightarrow \int \frac{d x}{x^{\frac{1}{3}}} \propto \int d t \\ \Rightarrow x^{\frac{2}{3}} \propto t \\ \Rightarrow x \propto t^{\frac{3}{2}} \end{array}$