Billiard Ball Collision with Wall
Understanding forces and impulses during collisions
Theory
Two identical billiard balls strike a rigid wall with the same speed but at different angles, and get reflected without any change in speed. We examine:
- The direction of the force on the wall due to each ball
- The ratio of the magnitudes of impulses imparted to the balls by the wall
Key Concepts:
1. The force on the wall is always normal (perpendicular) to the wall, regardless of the ball's approach angle.
2. Impulse equals change in momentum: \( \vec{J} = \Delta\vec{p} = m\Delta\vec{v} \)
3. Newton's Third Law: Force on wall = -Force on ball
Case (a) - Normal incidence:
Initial momentum: \( p_x = mu \), \( p_y = 0 \)
Final momentum: \( p_x = -mu \), \( p_y = 0 \)
Impulse: \( J_x = -2mu \), \( J_y = 0 \)
Case (b) - 30° incidence:
Initial momentum: \( p_x = mu\cos30° \), \( p_y = mu\sin30° \)
Final momentum: \( p_x = -mu\cos30° \), \( p_y = mu\sin30° \)
Impulse: \( J_x = -2mu\cos30° \), \( J_y = 0 \)
Impulse ratio: \( \frac{J_a}{J_b} = \frac{2mu}{2mu\cos30°} = \frac{1}{\cos30°} = \frac{2}{\sqrt{3}} \)
