Questions: Collision Analysis and Real-World Applications

Question 1

A cue ball strikes a stationary billiard ball off-center (a glancing blow). In what direction does the struck ball move immediately after impact?

• A. In the direction the cue ball was originally traveling
• B. Along the line connecting the two balls' centers at the moment of contact (the contact normal)
• C. Perpendicular to the cue ball's original direction of travel
• D. In the direction of the average of the cue ball's velocity and the contact line

Answer: (B)

In a collision between billiard balls, momentum is only exchanged along the line connecting the centers at the moment of contact — the contact normal. Momentum perpendicular to this line is unaffected. So the struck ball moves along the line of centers, regardless of the cue ball's original direction of travel. Option A is the most common misconception: students assume the struck ball inherits the cue ball's direction, but it's the collision geometry (line of centers), not the cue ball's trajectory, that governs the outcome.

Question 2

A crash reconstructionist arrives at a scene where two cars collided head-on and came to rest together. Kinetic energy was clearly not conserved — the cars are crumpled. Can she determine the pre-impact speeds?

• A. No — because kinetic energy was not conserved, the pre-impact velocity information is permanently lost
• B. Yes — she uses skid marks and friction to calculate the post-impact combined velocity, then applies momentum conservation to back-calculate pre-impact speeds
• C. No — crash reconstruction requires elastic collisions, which this was not
• D. Yes — but only by estimating the coefficient of restitution from crush depth alone

Answer: (B)

Momentum is always conserved in a collision, regardless of whether it is elastic or inelastic. The crash reconstructionist's method runs the physics backwards: use skid marks and friction coefficients to find the combined post-impact velocity, then use conservation of momentum (which holds exactly) to solve for the pre-impact velocities. The non-conservation of kinetic energy is irrelevant — momentum conservation provides enough equations. Option A reflects a common confusion: energy being lost doesn't mean momentum is lost.

Question 3

In a perfectly elastic head-on collision between two identical billiard balls, if one ball is initially stationary, the moving ball stops completely and the stationary ball moves off with the original velocity.

• T. True
• F. False

Answer: True

This is a classic result that follows directly from applying both conservation of momentum and conservation of kinetic energy to two equal-mass objects. The solution to the system of equations gives v₁' = 0 and v₂' = v₁ — the moving ball stops and the stationary one takes on the exact original velocity. This counterintuitive result is confirmed by experience with billiards and Newton's cradle.

Question 4

In a real inelastic collision, both kinetic energy and momentum are partially lost, so momentum conservation cannot be applied.

• T. True
• F. False

Answer: False

Momentum is always conserved in any collision — elastic, inelastic, or perfectly inelastic — as long as no external forces act during the collision. What is not conserved in inelastic collisions is kinetic energy, which converts to heat, sound, and deformation. The fact that kinetic energy is lost does not affect momentum conservation. This distinction is fundamental: crash reconstruction, forensic ballistics, and particle physics all rely on momentum conservation even in highly inelastic impacts.

Question 5

Why is the line of centers at the moment of impact — rather than the cue ball's direction of travel — the key geometric factor determining where a struck billiard ball goes?

Model answer: The contact force between the two balls acts only along the line connecting their centers at the moment of impact (the contact normal). This is the only direction in which the balls can push on each other — they cannot exert force perpendicular to the contact surface. Therefore, momentum is exchanged only along this line: the struck ball gains momentum in the direction of the line of centers, while momentum perpendicular to it remains unchanged. The cue ball's direction of travel determines how much of its momentum is directed along the contact normal, but the struck ball's resulting direction is always along that normal.

This is a key insight in 2D collision geometry: the contact geometry, not the incoming trajectory, determines the post-collision directions. This is why skilled billiards players aim based on the line of centers (the 'contact point') rather than simply shooting at the target ball's center — the angle of the line of centers at impact determines exactly where the struck ball will travel.