Question

A particle is moving with speed $v=b\sqrt{x}$ along positive $x$−axis. Calculate the speed of the particle at time $t=\tau$ (assume that the particle is at origin at $t=0$).

• A. $\frac{b^2\tau }{4}$
• B. $\frac{b^2\tau }{\sqrt{2}}$
• C. $b^2\tau$
• D. $\frac{b^2\tau }{2}$

Answer 1

The correct option is D $\frac{b^2\tau }{2}$

The speed of the particle, $v=b\sqrt{x}$
$\frac{dx}{dt}=b\sqrt{x}$
$\Rightarrow \frac{dx}{dt}=b{x}^{\frac{1}{2}}$

${\int }_0^x\frac{dx}{x\frac{1}{2}}={\int }_0^{t}bdt$

$2\sqrt{x}=bt$

$x=\frac{b^2{t}^2}{4}\Rightarrow v=\frac{dx}{dt}=\frac{b^{2}t}{2}$
Hence speed of the particle at time $t=\tau$ is
$\frac{b^2\tau }{2}$