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 $v=a\sqrt{x}$. The displacement of the particle varies with time as

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

Answer 1

The correct option is A $t^2$

Velocity, $v=a\sqrt{x}$

$\frac{dx}{dt}=a\sqrt{x}(\because v=\frac{dx}{dt})$

$\frac{dx}{\sqrt{x}}=adt$

Perform integration

${\int }_0^x\frac{dx}{\sqrt{x}}={\int }_0^{t}adt$

($\because$ at t=0, x=0 and let at any time t, particle is at x)

$\Rightarrow \frac{x^{1/2}}{1/2}{|}_0^x=at$

$\Rightarrow x^{1/2}=\frac{a}{2}t$

$\Rightarrow x=\frac{a^2}{4}\times t^2$

$\Rightarrow x\propto t^2$