Two particles collide directly; the rebounding one then hits a smooth fixed wall - find the restitution coefficient from whether a second collision happens
IAL M2 · Momentum, impulse and restitution · appears in 7 of 25 sittings, 2017-2024
Two particles collide directly on a smooth horizontal line, with a given or unknown coefficient of restitution between them. The particle that rebounds towards a smooth fixed vertical wall (perpendicular to the line of motion) hits it with its own coefficient of restitution, and rebounds again. Find the unknown coefficient of restitution (either e, between the particles, or f, at the wall) from the condition that a second collision between the two particles does, or does not, occur.
The setup itself - two particles collide, then one rebounds from a smooth fixed wall - recurs in 15 of 25 M2 sittings. The other 8 end on a different task: the time between the two collisions (2), where the second collision happens (1), the impulse in it (1), or the wall's coefficient from an energy, position or speed condition rather than from whether a second collision happens (4).
The method
5 marks at its longest. Each step below is a mark the examiner is looking for.
Find each particle's velocity after the first collision, using conservation of linear momentum and Newton's law of restitution (unknowns left in terms of e/u where e is not yet known) (2 marks)
Find the rebounding particle's velocity after it bounces off the wall, applying the wall's coefficient of restitution (1 mark)
Form the inequality (or, at the boundary, equation) comparing the rebounding particle's post-wall velocity against the other particle's velocity, matching the stated second-collision condition; the inequality must be dimensionally correct (1 mark)
Solve the inequality for the unknown coefficient and state the resulting range, including both endpoints where the question requires them (1 mark)
What changes between years
Everything below varies from sitting to sitting. The method above does not.
which coefficient is the unknown: e (2022-January) or f (2023-October, 2024-January)
the condition to apply: "a second collision occurs" (2022-January, 2024-January) vs "no second collision occurs" (2023-October) - the inequality's direction flips accordingly
the particle masses and initial-speed ratios differ in every sitting (2m/3m at u/2u; 4m/2m at 2u/3u; m/5m)
whether the two particles are moving in the same or opposite directions before the first collision (opposite in 2022-January and 2023-October; same direction in 2024-January)
Where the marks go
Instructions the examiners are given, repeated across sittings.
The final range must state both limits where both are asked for - e.g. 2023-October's mark scheme requires "0 <= f <= 5/16, both ends required" (2 sittings)
The second-collision inequality is a dependent method mark, only awarded once the wall-rebound velocity step has already scored (1 sittings)
Strict vs non-strict inequality matters and is checked explicitly against the physical condition stated ("second collision occurs" needs a strict inequality; "no collision" allows equality) (2 sittings)
GOAL-15 Phase 3 recounted this over all 25 M2 sittings (it had been counted over 11). The headline is the exact closing step: a coefficient, or its range, from whether a second collision occurs. 2017-June is the nearest miss: the same closing step, but against a third particle with no wall; 2016-October is a weaker one (three particles, shows no second collision for given values).