Question

A rock of mass $m$ slides down with an initial velocity $v_0$. A retarding force $F=-k\sqrt{v}$ acts on the body. The velocity at any instant is given by:

• A. $v=v_0-\frac{kt}{m}$
• B. $v=v_0-{\left(\frac{kt}{m}\right)}^2$
• C. $\sqrt{v}=\sqrt{v_0}-\frac{kt}{2m}$
• D. none of these

Answer 1

The correct option is C $\sqrt{v}=\sqrt{v_0}-\frac{kt}{2m}$

$F=-k{v}^{1/2}$

$m\frac{dv}{dt}=-k{v}^{1/2}$

$v^{-1/2}dv=-\frac{k}{m}dt$

Integrating,

$2\sqrt{v}=-\frac{kt}{m}+c$ where $c=$ integration constant.

at $t=0,v=v_0,so,c=2\sqrt{v_0}$

now,

$\sqrt{v}=\sqrt{v_0}-\frac{kt}{2m}$