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Learning objectives
Core idea
Key formulae
Worked examples
Explore the graph
Common mistakes
Exam tip
Guided practice
Practice questions
Challenge
Summary
Quick quiz
Same topic, different board
Past paper questions

Learn9FM0Option · Further Mechanics 1Elastic collisions in one dimensionNewton's law of restitution

Elastic collisions in one dimension · Subtopic 1 of 1

Collisions in One Dimension

Momentum is conserved in a collision, but energy usually isn't. Newton's law of restitution adds the missing equation: speed of separation $=e\times$ speed of approach, with $0\le e\le1$ . For a wall, rebound speed $=e\times$ impact speed.

Edexcel 9FM0 Option≈ 45 minRevision 1 · updated 2026-10-27

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Section 1 of 14

Learning objectives

Use $v_2-v_1=e(u_1-u_2)$ with conservation of momentum.
Solve impacts with a fixed wall.
Find kinetic energy lost and analyse successive impacts.

Pause and think

Drop a tennis ball and a lump of clay from the same height. Both hit the floor at the same speed. What's different?

Section 2 of 14

Core idea

Start hereBouncy and dead

Drop a tennis ball and a lump of clay from the same height. They hit the floor at the same speed, but only one bounces back. Momentum alone can't tell you what happens next; we need a measure of bounciness.

The big idea

In a direct collision, momentum is conserved and Newton's law of restitution gives a second equation: speed of separation $=e\times$ speed of approach. Two equations, two unknown final velocities.

1. The two equations

m_1u_1+m_2u_2=m_1v_1+m_2v_2,\qquad v_2-v_1=e(u_1-u_2)

$e$: coefficient of restitution, $0\le e\le1$ .
$u,v$: velocities before and after, all positive in the same direction.

Worked exampleSphere hits sphere

A ( $2$ kg, $5$ m s $^{-1}$ ) hits B ( $3$ kg, at rest) with $e=\frac12$ . Find both velocities after.

Goal: $v_A,v_B$ .
Strategy: Momentum and restitution, solve simultaneously.

$10=2v_A+3v_B$
Momentum.
$v_B-v_A=\tfrac12(5-0)=2.5$
Restitution.
$10=2(v_B-2.5)+3v_B\Rightarrow v_B=3$
Substitute $v_A=v_B-2.5$ .
$v_A=0.5$
Back-substitute.

Answer

v_A=0.5

v_B=3

m s

^{-1}

(both forwards).

✓ CheckMomentum

1+9=10

✓; separation

2.5=\frac12\times5

✓.

2. Energy loss and walls

Ball dropped from height

h

$u=\sqrt{2gh}$
Speed on reaching the floor.
$v=eu$
Restitution with a fixed surface.
$h_1=\frac{v^2}{2g}=e^2h$
Height reached on the bounce.

Each bounce multiplies the height by

e^2

Worked exampleKinetic energy lost

Find the KE lost in the sphere collision above.

Goal: Loss in J.
Strategy: Before minus after.

$\tfrac12(2)(5^2)=25$
KE before.
$\tfrac12(2)(0.5^2)+\tfrac12(3)(3^2)=0.25+13.5$
KE after.

Answer

11.25

J lost.

✓ Check

e<1

so a loss is expected ✓.

Solving a direct impact

Drawing diagram…

The two equations work together every time.

Common misconceptionOrder of subtraction Separation is

v_2-v_1

(front minus back) and approach is

u_1-u_2

(back minus front). Swapping one but not the other flips the sign.

Key takeawayLimits of e

e=1

: perfectly elastic, KE conserved.

e=0

: the bodies coalesce and move together.

Pause and think

Equal masses, one at rest, $e=1$ . What happens?

Summary

Momentum + restitution.
Wall: $v=eu$ .
KE lost unless $e=1$ .

Coefficient of restitution: Ratio of separation speed to approach speed.
Direct impact: Velocities along the line of centres.

HistoryNewton. Newton measured restitution with pendulum balls of wool, steel and glass, and found the ratio of speeds depended only on the materials.

Fun factBall inspection. Wimbledon tennis balls must bounce between 53% and 58% of their drop height (about 135–147 cm from 254 cm): a test of

e^2

Section 3 of 14

Key formulae

Momentum

m_1u_1+m_2u_2=m_1v_1+m_2v_2

Restitution

v_2-v_1=e(u_1-u_2)

Wall

v=eu

TrickArrows the same way. Draw all velocities positive to the right, before and after. Let the algebra give negative answers for leftward motion.

Section 4 of 14

Worked examples

Example 1 · Two spheres0 / 3 steps

A (

2

kg,

5

m s

^{-1}

) hits B (

3

kg, at rest);

e=\frac12

. Find the velocities after.

Example 2 · Energy loss0 / 2 steps

Find the KE lost in Example 1.

Section 5 of 14

Explore the graph

Speeds after impact as e varies

Loading the interactive graph…

Try this

$x$ is $e$ . Lower line: A's velocity $2-3e$ ; upper line: B's velocity $2+2e$ .
At $e=0$ they move together at $2$ m s $^{-1}$ ; for $e>\frac23$ , A bounces backwards.

Momentum fixed; restitution decides how it's shared.

In the real worldCrash testing. Car crumple zones make

e

small so the car absorbs energy rather than bouncing the occupants back.

Section 6 of 14

Common mistakes

Incorrect

v_1-v_2=e(u_1-u_2)

Correct

v_2-v_1=e(u_1-u_2)

Separation is the front one minus the back one.

IncorrectConserving kinetic energy

CorrectOnly if

e=1

Most collisions lose energy.

IncorrectIgnoring signs

CorrectConsistent positive direction

Opposite motion is negative.

Kinetic energy: conserved?

Restitution: only when e = 1, mate.

Section 7 of 14

Exam tip

Exam tipSecond collision test. After hitting a wall, a sphere collides again only if it then moves towards the other one faster than it is moving away.

Section 8 of 14

Guided practice

SA-FM-C1-G011 markL1 FoundationNon-calculatorWalls

A ball hits a wall at

8

m s

^{-1}

with

e=0.25

. Find its rebound speed.

Salam Academy Original Practice · Proposed mark scheme

Section 9 of 14

Practice questions

SA-FM-C1-0013 marksL2 StandardNon-calculatorRestitution

Sphere A (

2

kg,

5

m s

^{-1}

) hits sphere B (

3

kg, at rest) directly with

e=\frac12

. Find B's speed afterwards.

Salam Academy Original Practice · Proposed mark scheme

SA-FM-C1-0022 marksL2 StandardNon-calculatorEnergy loss

In the last question, find the kinetic energy lost in joules.

Salam Academy Original Practice · Proposed mark scheme

SA-FM-C1-0032 marksL2 StandardNon-calculatorWalls

A ball is dropped and hits the floor at

10

m s

^{-1}

, rebounding at

6

m s

^{-1}

. Find

e

Salam Academy Original Practice · Proposed mark scheme

SA-FM-C1-0043 marksL2 StandardNon-calculatorRestitution

Two identical spheres collide head-on, each at

4

m s

^{-1}

, with

e=\frac12

. Find the speed of each afterwards.

Salam Academy Original Practice · Proposed mark scheme

More questions on every skill in this subtopic are in the question bank under Practise.

Section 10 of 14

Challenge

SA-FM-C1-0053 marksL4 ChallengingNon-calculatorEnergy loss

A ball is dropped from

5

m onto a floor with

e=\frac12

. Find the height of the first bounce in metres.

Salam Academy Original Practice · Proposed mark scheme

Section 11 of 14

Summary

Momentum + restitution.
$0\le e\le1$ ; $e=1$ perfectly elastic.
Wall: $v=eu$ .

Section 12 of 14

Quick quiz

Q1 · SA-FM-C1-Q011 markL1 FoundationNon-calculatorRestitution

A perfectly elastic collision has

Salam Academy Original Practice · Proposed mark scheme

Section 13 of 14

Same topic, different board

Specification: A level Further Mathematics 9FM0
Where: Further Mechanics 1 · Elastic collisions in one dimension
Expected: Newton's law of restitution, direct impact of spheres, impact with a fixed surface, loss of kinetic energy, successive impacts.
Calculator: Permitted; show enough working to earn the method marks.
Typical style: Edexcel often opens with a routine part and builds to a "show that" or "prove" part.
Open in: 9FM0 · Option

Section 14 of 14

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Salam Academy original content · Content status: reviewed and published. Syllabus mapping: Pearson Edexcel 9FM0, First teaching 2017 (official specification).