Question

A particle moves with a deceleration $k\sqrt{v}$. Initial velocity is $v_0$. Find the time after which it will stop.

• A. $\frac{\sqrt{v_0}}{k}$
• B. $\frac{\sqrt{v_0}}{2k}$
• C. $\frac{2\sqrt{v_0}}{k}$
• D. none of these

Answer 1

The correct option is C $\frac{2\sqrt{v_0}}{k}$

Given $a=-k\sqrt{v}$

Thus, $\frac{dv}{dt}=-k\sqrt{v}or,{\int }_{v_0}^0{v}^{-1/2}dv=-k{\int }_0^{t}dt\Rightarrow {2\left(v^{1/2}\right)}_{v_0}^0=-kt$

$⟹0-2\sqrt{v_0}=-kt$

Thus, $t=\frac{2\sqrt{v_0}}{k}$