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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 $t^{n} then n =$

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answer is 2.

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Detailed Solution

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The variation of the velocity is given as $= α \sqrt{x}$

We can write it as $\frac{dx}{dt} = α \sqrt{x} (∵ v = \frac{dx}{dt}) \Rightarrow \frac{dx}{\sqrt{x}} = α dt$ Perform integration with the limit $\int_{0}^{x} \frac{dx}{\sqrt{x}} = \int_{0}^{t} αdt \Rightarrow \frac{x^{1 / 2}}{1 / 2}_{0}^{x} = αt \Rightarrow x^{1 / 2} = \frac{α}{2} t \Rightarrow x = \frac{α^{2}}{4} \times t^{2} \Rightarrow x \propto t^{2}$

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