Question

A ball of mass $m$ is moving with a velocity $v$ towards a rigid vertical wall. After striking the wall, the ball gets deflected by an angle $\theta$ without any change in its speed. The impulse imparted to the ball will be

• A. $2v\cos \left(\theta \right)$
• B. $2v\cos \left(\frac{\theta }{2}\right)$
• C. $2v\sin \left(\frac{\theta }{2}\right)$
• D. $2v\sin \left(\theta \right)$

Answer 1

The correct option is B $2v\cos \left(\frac{\theta }{2}\right)$

Let $v_1$ and $v_2$be the initial and final velocities of the ball as shown in fig.(a)
Impulse = change in momentum
$=m{v}_2-m{v}_1=m(v_2-v_1)=m[v_2+(-v_1)]=m\Delta v$
where $\Delta v=v_2+(-v_1)$is the resultant of $v_2$and $-v_1$as shown in fig.(b)
Magnitude of $\Delta v$ is
[magnitude of $v_1$= magnitude of $v_2$=$v$]
$\Delta v=\sqrt{{v_1}^2+{v_2}^2+2v_1{v}_2\cos \left(\theta \right)}=\sqrt{v^2+v^2+2v^2\cos \left(\theta \right)}=\sqrt{2v^2(1+\cos (\theta \left)\right)}=2v\cos \left(\frac{\theta }{2}\right)$
The direction of impulse is perpendicular to the wall and away from it.