Question

A force $F=kt(\tau -t)$ acts on a particle of mass $m$, which is at rest at $t=0$ where $k$ is a constant. Find the momentum of the force when the action of the force is discontinued

• A. $k{\tau }^3/2$
• B. $k{\tau }^3/3$
• C. $k{\tau }^3/6$
• D. $k{\tau }^3/4$

Answer 1

Answer: (C)

$k{\tau }^3/6$

$F=ma=kt(\tau -t)$
$m\frac{dv}{dt}=kt(\tau -t)$
integrating, $v=\frac{k}{m}[\tau \frac{t^2}{2}-\frac{t^3}{3}]+c$ , where $c=$ integrating constant.
at $t=0,v=0$, thus $c=0$
$v=\frac{k}{m}[\tau \frac{t^2}{2}-\frac{t^3}{3}]$
the action of force is discontinued when $t=\tau$
so, $v=\frac{k}{m}[\frac{{\tau }^3}{2}-\frac{{\tau }^3}{3}]=\frac{k{\tau }^3}{6m}$
The momentum , $mv=\frac{k{\tau }^3}{6}$