Example

Question:

Two identical billiard balls strike a rigid wall with the same speed but at different angles, and get reflected without any change of speed, as shown in Fig. 4.6.
What is (i) the direction of the force on the wall due to each ball?
(ii) the ratio of magnitudes of impulses imparted to the balls by the wall?

Solution:

Case (a):
\[ (p_x)_{\text{initial}} = mu \quad (p_y)_{\text{initial}} = 0 \] \[ (p_x)_{\text{final}} = -mu \quad (p_y)_{\text{final}} = 0 \] Impulse is the change in momentum vector.
Therefore,
x-component of impulse \( = -2mu \)
y-component of impulse \( = 0 \)
Impulse and force are in the same direction.
Thus, the force on the ball due to the wall is normal to the wall, along the negative x-direction.

Case (b):
\[ (p_x)_{\text{initial}} = mu\cos 30^\circ,\quad (p_y)_{\text{initial}} = -mu\sin 30^\circ \] \[ (p_x)_{\text{final}} = -mu\cos 30^\circ,\quad (p_y)_{\text{final}} = -mu\sin 30^\circ \] Note, while \(p_x\) changes sign after collision, \(p_y\) does not.
Therefore,
x-component of impulse \( = -2mu\cos 30^\circ \)
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.
By Newton's third law, the force on the wall due to the ball is normal to the wall along the positive x direction.

Ratio of the magnitudes of the impulses imparted to the balls:
\[ \frac{2mu}{2mu\cos 30^\circ} = \frac{2}{\sqrt{3}} \approx 1.2 \]