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

Given velocity as a function of ‘x’ 

i.e  $v=b\sqrt{x}$

diff on both sides with respect to t

We get  $\therefore \frac{dv}{dt}=\frac{b}{2\sqrt{x}}\times \frac{dx}{dt}=\frac{bv}{2\sqrt{x}}=\frac{b\left(b\sqrt{x}\right)}{2\sqrt{x}}=\frac{b^2}{2}$

$a=\frac{b^2}{2}\Rightarrow \frac{dv}{dt}=\frac{b^2}{2}\Rightarrow dv=\frac{b^2}{2}\times dt$

Integrating on both sides, $v=\frac{b^2}{2}\times t=\frac{b^2}{2}\times \tau$