Two identical billiard balls striks a rigid wall with the same speed but at different angles , and get reflected without any change in speed , as shown in Fig . What is (i) the direction of the force on the wall due to each ball ? (ii) the ratio of the magnitudes of impulses imparted to the balls by the wall ?

Explanation

An instinctive answer to (i) might be that the force on the wall in case (a) is normal to the wall, while that in case (b) is inclined at `30^(@)` to the normal. This answer is wrong. The force on the wall is normal to the wall in both cases .
How to find the force on the wall ? The trick is to consider the force ( or impulse ) on the ball due to the wall using the second law , and then use the third law to answer (i) . Let u be the speed of each ball before and after collision with the wall, and m the mass of each ball . Choose the x and y axes as shown in the figure, and consider the change in momentum of the ball in each case :
Case (a)
`(p_x)_initial` = mu `(p_y)_initial` = 0
`(p_x)_final` = -mu `(p_y)_final` = 0
Impulse is the change in momentum vector.
Therefore,
x-component of impulse = -2 mu
y-component of impulse = o
Impulse and force are in the same direction. Clearly, fro above , the force on the ball due to the wall is normal to the wall , along the negative x-direction. Using Newton's third law of motion, the force on the wall due to the ball is normal to the wall along the positive x-direction .The magnitude of force cannot be ascertained since the small time taken for the collision has not been specified in the problem.
Case (b)
`(p_x)_initial` = mucos`30^(@)` , `(p_y)_initial` = -`musin30^(@)`
`(p_x)_final` = `-mucos30^(@)` , `(p_y)_final` = `- musin30^(@)`
Note, while `p_x`, changes sign after collision , `p_y` does not. Therefore,
x-component of impulse = `- 2 m u cos 30^(@)`
y-component of impulse = 0
The direction of impulse (and force ) is the same as in (a) and is normal to the wall along the negative x direction . As before , using Newton's third law, the force on the wall due to the ball is normal to the wall along the positive x direction .
The ratio of the magnitudes of the impulses imparted to the balls in (a) and (b) is
2 m u / (2 m u `cos 30^(@)` ) = 2 / `sqrt 3` = 1.2