Question

A sphere of mass $m$ slides with velocity $v$ on a frictionless surface towards a smooth inclined wall as shown in figure. If the sphere returns with the same speed after collision with the wall, then the impulse imparted by the inclined wall on the sphere will be:

• A. $2mv\text{cosec}\theta$
• B. $2mv\sin \theta$
• C. $2mv\cot \theta$
  
• D. $2mv\tan \theta$

Answer 1

The correct option is A $2mv\text{cosec}\theta$


Let $J_1=$ Impulse imparted to ball by the inclined wall
$J_2=$ Impulse imparted to ball by the horizontal surface
$\because$ the ball is not bouncing in vertical direction after impact
$J_{net}$ in $y$ direction $=0$
$\Rightarrow J_2=J_1\cos \theta ...\left(i\right)$

From the figure showing components of impulse on the ball:


$\text{Net impulse in x direction =}$
$\text{change in momentum in x-direction}$
$J_1\sin \theta =m{v}_f-m{v}_i$
$\Rightarrow J_1\sin \theta =(+mv)-(-mv)$
(taking +ve $x$ axis rightward)
$\therefore J_1=\frac{2mv}{\sin \theta }=2mv\text{cosec}\theta$