Question

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

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

Answer 1

Answer: (A)

$v=\alpha \sqrt{x}$
$\frac{dx}{dt}=\alpha \sqrt{x}\left(\because v=\frac{dx}{dt}\right)$
$\frac{dx}{\sqrt{x}}=\alpha dt$
Perform integration
${\int }_0^x\frac{dx}{\sqrt{x}}={\int }_0^i\alpha dt$
[$\because$ at $t=0,x=0$ and let at any time $t$, particle is at $x$]
$\Rightarrow \frac{x^{-1/2}}{-1/2}{|}_0^x=\alpha t$
$\Rightarrow x^{-1/2}=\frac{\alpha }{2}t$
$\Rightarrow x=\frac{{\alpha }^2}{4}\times t^2\Rightarrow x\propto t^2$