Two identical billiard balls strike a rigid wall with the same speed but at different angles and get reflected without any loss of speed as shown in the given figure. What is the ratio of the magnitudes of the impulse imparted on the two balls by the wall?

Resolve the velocity of the ball along $x and y-axis$, we have

$$\begin{gathered} \overrightarrow{u_x } = u \cos 30° \hat{i} ; \overrightarrow{u_y } = – u \sin 30° \hat{j} \\ and, \overrightarrow{v_x } = – u \cos 30° \hat{i}; \overrightarrow{v_y } = – u \sin 30° \hat{j} \\ The impulse exerted by the wall on the ball; \\ \overrightarrow{J_x}= m \left(v_x - u_x\right) = m (– u \cos 30° \hat{i} – u \cos 30° \hat{i}) \\ = – 2mu \cos 30° \hat{i}= 3 mu \hat{i} \\ \overrightarrow{J_y}= m \left(v_y - uy\right) = m (– u \sin 30° \hat{j} + u \sin 30° \hat{j}) = 0 \\ Resul\tan t impulse, \\ \overrightarrow{J }=\overrightarrow{J_x }+\overrightarrow{J_y} = – \sqrt{3 }mu \widehat{i } \end{gathered}$$

The force exerted by wall on ball is along negative x-axis. By Newton’s third law the direction of force on wall by ball is along positive $x-axis.$ The ratio of impulse imparted by the wall on balls

$\frac{J_1}{J_2} = \frac{2}{\sqrt{3}}$