A particle located at x=0 at time t = 0, starts moving along the positive x-direction with a velocity V that varies as $V = α \sqrt{x}$ . The displacement of the particle varies with time as t^x then x =

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Detailed Solution

The variation of the velocity is given as

$V = α \sqrt{x}$

We can write it as $\frac{dx}{dt} = α \sqrt{x} (∵ v = \frac{dx}{dt})$

$\Rightarrow \frac{dx}{\sqrt{x}} = αdt$

Perform integration with the limit $\int_{0}^{x} \frac{dx}{\sqrt{x}} = \int_{0}^{t} αdt$

$\begin{array}{l} {\Rightarrow \frac{x^{1 / 2}}{1 / 2}|}_{0}^{x} = αt \Rightarrow x^{1 / 2} = \frac{α}{2} t \\ \Rightarrow x = \frac{α^{2}}{4} \times t^{2} \\ \Rightarrow x \propto t^{2} \end{array}$

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