Opposite Directions and Several Collisions

Head-on collisions and rebounds are where signs matter. Fix a positive direction, give every velocity a sign, and let the algebra tell you which way things end up moving.

Cambridge 9709 Paper 4Cambridge 9231 Paper 3Edexcel YMA01 WME02Edexcel 9FM0 OptionAQA 7367 Paper 3≈ 35 minRevision 5 · updated 2026-10-27

Section 1 of 14

Learning objectives

Use signed velocities for particles moving towards each other.
Interpret a negative answer as motion in the negative direction.
Work through a sequence of collisions one at a time.
Decide whether a further collision will happen.

Pause and think

A $2$ kg trolley at $3\ \text{m s}^{-1}$ and a $3$ kg trolley at $2\ \text{m s}^{-1}$ roll towards each other and stick. Which way do they move afterwards?

Section 2 of 14

Core idea

Start hereMinus signs mean direction

Two dodgem cars meet head-on. One moves right, one left. Momentum is a vector, so direction matters: choose one direction as positive and give everything moving the other way a minus sign. A negative answer is information, not an error.

The big idea

Choose a positive direction. Velocities the other way are negative. Solve the momentum equation; a negative result means motion in the negative direction.

1. Head-on

Head-on: one velocity is negative

Before:

2(4)+3(-3)=-1

. After:

2v_1+3(2)=-1

, so

v_1=-3.5

: the 2 kg ball bounces back at

3.5

m/s.

Worked example2 kg at 4 → meets 3 kg at 3 ←

After, the 3 kg moves → at 2 m/s. Find the 2 kg's velocity.

Goal: $v_1$ .
Strategy: Right positive.

$2(4)+3(-3)=-1$
Before.
$2v_1+3(2)=-1\Rightarrow v_1=-3.5$
After.

Answer

3.5

m/s to the left (it bounces back).

✓ Check

2(-3.5)+6=-1

✓.

Worked exampleA chain of collisions

A (1 kg, 6 m/s) hits B (2 kg, at rest) and stops; B hits C (4 kg, at rest) and stops. Speed of C?

Goal: $v_C$ .
Strategy: Two separate collisions.

$6=2v_B\Rightarrow v_B=3$
First.
$2(3)=4v_C\Rightarrow v_C=1.5$
Second.

Answer

1.5

m/s.

✓ CheckMomentum

6

passed along unchanged ✓.

Exam tipDiagrams Draw before and after pictures with masses, velocities and the positive direction.

Pause and think

Equal masses, equal speeds, head-on, stick. Final speed?

Summary

Positive direction; signs.
Negative answer = opposite way.

Vector: Has size and direction.

HistoryWren's pendulums. To test their collision laws in 1668, Christopher Wren (later the architect of St Paul's Cathedral) hung balls on strings and let them swing into each other. The height a ball swung up to afterwards told him its speed. Head-on collisions with balls moving in opposite directions were the key test of the signed rule.

Fun factThe snooker stop shot. When a snooker cue ball hits an equal-mass ball head-on without spin, the cue ball stops dead and the object ball takes all its velocity. Professionals use this "stop shot" constantly. It is a nearly elastic collision between equal masses, and momentum and energy are both (almost) conserved.

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