Question

A particle starts with velocity $v_0$ at time $t = 0$ and is decelerated at a rate proportional to the square root of its speed at time t with constant of proportionality $\alpha$. The total time for which it will move before coming to rest is

• A. $\sqrt{v_{0}}$
• B. $\frac{v_{0}^{3 / 2}}{\alpha}$
• C. $\frac{2 v_{0}^{3 / 2}}{\alpha}$
• D. $\frac{2 \sqrt{v_{0}}}{\alpha}$

Answer 1

Answer: (D)

$u = v _{0}, \quad a =-\alpha \sqrt{ v }$
so, $\frac{d v}{d t}=-\alpha \sqrt{v}$
$\Rightarrow \int_{v_{0}}^{0} \frac{1}{\sqrt{v}} d v=\int_{0}^{t}-\alpha d t$
$\Rightarrow t =\frac{2 \sqrt{v_{0}}}{\alpha}$