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

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Solution

Q.
A body is moving unidirectionally under the influence of a constant power source. Its displacement in time $t$ is proportional to [2024]

1 $t$

2 $t^{2}$

3 $t^{2 / 3}$

4 $t^{3 / 2}$

Ans.
(4)

$\frac{d W}{d t} = P \Rightarrow dW = P d t$

   $F \cdot d s = P d t$

$m d v \frac{d s}{d t} = P d t$

   $\int_{0}^{v} v d v = \frac{P}{m} \int_{0}^{t} d t \Rightarrow \frac{v^{2}}{2} = \frac{P t}{m}$

Hence,   $v \propto \sqrt{t} \Rightarrow \frac{d s}{d t} = C \sqrt{t},$ $C$ is a constant

$\int_{0}^{s} d s = C \int_{0}^{t} t^{1 / 2} d t$

   $s = C \frac{2}{3} t^{3 / 2} \Rightarrow s \propto t^{3 / 2}$

$∴$   displacement of proportional to ${(t)}^{3 / 2}$

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