1. Impulse and Momentum
The linear momentum of a particle is the product of its mass and velocity. It is a vector quantity.
When force varies with time, impulse = ∫F dt. For a constant force over time t: impulse = Ft. Either way, impulse = change in momentum.
Conservation of Linear Momentum
When no external forces act on a system, the total linear momentum is conserved. This is the most important principle in collision problems.
Always holds — regardless of the type of collision (elastic, inelastic, or perfectly inelastic).
2. Newton's Experimental Law — Coefficient of Restitution
Newton's experimental law states that the relative speed of separation after impact equals e times the relative speed of approach before impact, where e is the coefficient of restitution.
= (v₂ − v₁) / (u₁ − u₂) [along line of centres]
Always measured along the line of centres (the line joining the centres of the two spheres at impact).
The Coefficient of Restitution e
(bodies coalesce) 0<e<1: Inelastic
(KE lost) e=1: Perfectly elastic
(KE conserved)
Always define a positive direction before writing equations. Velocities in the opposite direction are negative. The restitution equation uses:
3. Direct Impact — Two Spheres
A direct impact (head-on collision) occurs when the velocities of both particles are along the line of centres. We have two unknowns (v₁ and v₂) and two equations.
Direct collision: velocities along line of centres.
Two equations, two unknowns:
Solve simultaneously. Always check: 0 ≤ e ≤ 1 and no sphere passes through the other (v₁ ≤ v₂ if A was overtaking B).
4. Impact with a Fixed Surface
When a sphere hits a fixed smooth surface, the component of velocity perpendicular to the surface is reversed and scaled by e. The component parallel to the surface is unchanged (smooth surface).
For a direct impact on a wall (velocity perpendicular to wall): v = eu. The wall exerts an impulse J = m(v + u) = m(e+1)u on the sphere.
5. Oblique Impact — The Key 9231 Topic
In an oblique impact, the initial velocities are not along the line of centres. This is the most frequently examined and most challenging part of the momentum topic.
- Step 1 — Set up axes: Take the x-axis along the line of centres, y-axis perpendicular to it.
- Step 2 — Resolve velocities of each sphere into x and y components before impact.
- Step 3 — Along line of centres (x): Apply conservation of momentum AND Newton's restitution law → find v₁ₓ and v₂ₓ.
- Step 4 — Perpendicular to line of centres (y): Smooth spheres exert no force in this direction → each sphere's y-component is unchanged.
- Step 5 — Reconstruct final velocity vectors and find speed/angle if required.
Oblique impact: resolve into x (line of centres) and y (perpendicular). Only x-components change.
6. Energy Loss in Collisions
Kinetic energy is conserved only in a perfectly elastic collision (e = 1). For all other collisions (0 ≤ e < 1), KE is lost as heat, sound, and deformation.
When e = 1, ΔKE = 0 (elastic). When e = 0, ΔKE is maximum (perfectly inelastic — maximum energy loss). Energy lost ∝ (1 − e²) × (approach speed)².
Check: v₂ > v₁ ✓ (B faster than A — they don't overlap)
Angle to wall = 30°, so angle to normal = 60°.
The angle to the wall decreases from 30° to 19.1° because the normal component is reduced by e while the tangential component is unchanged.
Interactive Collision Laboratory
Set masses, initial velocities and the coefficient of restitution, then watch the direct collision play out. See post-collision velocities and energy loss computed in real time.
Practice Questions
Hmm — let's recheck. Using (1+3e²)/4: |v_A|² = u²(1+3e²)/4 requires v_Ax = u(1−e)/4, v_Ay = u√3/2:
This identity only holds for specific e. The correct expression for speed of A is u√[(13−2e+e²)/16]. Direction: tan θ = v_Ay/v_Ax = (u√3/2)/[u(1−e)/4] = 2√3/(1−e) from the line of centres.
Formula Reference Sheet
Complete reference for Momentum — Cambridge 9231 P3, Section 3.6.
- Always define positive direction at the start and mark it on your diagram. All velocities must be consistent with this choice.
- Write out both CLM and NEL clearly before solving — examiners expect to see both equations stated.
- After solving, check the physics: if two objects approach (A overtaking B), ensure v₁ ≤ v₂ after (A no longer overtaking). If not, something is wrong.
- For oblique impacts, resolve along and perpendicular to the line of centres — the y-component is unchanged for smooth spheres. This is the key exam distinction.
- For a ball bouncing on a floor, the coefficient of restitution gives
e = √(h₁/H)— a quick check available in successive bounce questions. - KE loss questions: always compute KE before and after separately. Don't use the formula unless you've derived it — show working.