Question

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

• A. $t^3$
• B. $t^2$
• C. $t$
• D. $t^{1/2}$

Answer 1

The correct option is B $t^2$

Given:

$\frac{dx}{dt}=\alpha \sqrt{x}$

$\Rightarrow \int \frac{dx}{\sqrt{x}}=\int \alpha$ dt

$\Rightarrow \frac{-1}{2}{x}^{1/2}=\alpha t$
$\Rightarrow x\propto t^2$