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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 two dimensionsOblique impacts

Elastic collisions in two dimensions · Subtopic 1 of 1

Oblique Collisions

For smooth bodies the impulse acts only along the line of centres (or the normal to a wall). So the velocity component parallel to the surface is unchanged, and the component along the normal obeys momentum and restitution exactly as in one dimension.

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

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Section 1 of 14

Learning objectives

Split velocities into components along and perpendicular to the line of impact.
Solve oblique impacts with a smooth plane.
Solve oblique impacts between two smooth spheres.

Pause and think

A snooker ball hits a cushion at an angle. Why does it leave at a shallower angle to the cushion than it arrived?

Section 2 of 14

Core idea

Start hereOff the cushion

A snooker ball hits a cushion at an angle and comes off at a different angle. During the very short impact, the cushion can only push straight out; it can't push sideways. That single fact is the whole method.

The big idea

For smooth bodies the impulse acts along the common normal (line of centres). Velocity components perpendicular to that line are unchanged; components along it obey momentum and restitution as in 1D.

1. Sphere and wall

Rebound angle

$v\cos\beta=u\cos\alpha$
Parallel component unchanged (angles measured from the wall).
$v\sin\beta=eu\sin\alpha$
Normal component reversed and multiplied by $e$ .
$\tan\beta=e\tan\alpha$
Divide the second by the first.

Since

e\le1

\beta\le\alpha

: the ball leaves closer to the wall.

Worked exampleWall impact

A smooth ball hits a smooth wall at $10$ m s $^{-1}$ , $60^\circ$ to the wall, $e=\frac13$ . Find the rebound speed and angle.

Goal: $v$ , $\beta$ .
Strategy: Components.

$10\cos60^\circ=5$
Parallel, unchanged.
$\tfrac13\times10\sin60^\circ=\tfrac{5\sqrt3}3$
Normal after impact.
$v=\sqrt{25+\tfrac{25}3}=\tfrac{10}{\sqrt3}$
Pythagoras.
$\tan\beta=\tfrac{\sqrt3}3\Rightarrow\beta=30^\circ$
Ratio of components.

Answer

5.77

m s

^{-1}

30^\circ

to the wall.

✓ Check

\tan\beta=e\tan\alpha=\frac13\sqrt3

✓.

2. Two spheres

Worked exampleGlancing collision

A moves at $3\mathbf i+4\mathbf j$ and hits identical B at rest; line of centres along $\mathbf i$ ; $e=1$ . Find both velocities.

Goal: $\mathbf v_A,\mathbf v_B$ .
Strategy: 1D along $\mathbf i$ ; keep $\mathbf j$ parts.

$v_A+v_B=3,\ v_B-v_A=3$
Momentum and restitution along $\mathbf i$ .
$v_B=3,\ v_A=0$
Solve.
$\mathbf v_A=4\mathbf j,\ \mathbf v_B=3\mathbf i$
$\mathbf j$ -components unchanged ( $4$ and $0$ ).

AnswerA:

4\mathbf j

; B:

3\mathbf i

✓ CheckThey move at right angles; KE

\frac12(25)=\frac12(16+9)

✓.

Method

Drawing diagram…

Split, solve along the line, recombine.

Common misconceptionWhich angle? Check whether the angle is given to the wall or to the normal; the sin and cos swap.

TrickVector form With the line of centres along unit vector

\hat{\mathbf n}

, the normal component of

\mathbf u

\mathbf u\cdot\hat{\mathbf n}

Pause and think

What happens to a ball hitting a wall head-on ( $\alpha=90^\circ$ )?

Summary

Smooth: impulse along the normal only.
Perpendicular components unchanged.
$\tan\beta=e\tan\alpha$ for a wall.

Oblique impact: Collision not along the line of motion.
Line of centres: Line through both centres at impact.

HistoryBilliards. Coriolis (of the Coriolis effect) wrote a mathematical theory of billiards in 1835, including oblique impacts and spin.

Fun factThe 90° rule. When equal smooth balls collide elastically and one is at rest, they leave at right angles: snooker players use this to plan the cue ball's path.

Section 3 of 14

Key formulae

Wall: parallel

u\cos\alpha=v\cos\beta

Wall: normal

v\sin\beta=eu\sin\alpha

Angle

\tan\beta=e\tan\alpha

TrickLine of centres. For two spheres, set axes along and perpendicular to the line joining their centres at impact. Do 1D work along it; leave the other components alone.

Section 4 of 14

Worked examples

Example 1 · Wall0 / 3 steps

A ball hits a smooth wall at

10

m s

^{-1}

60^\circ

to the wall, with

e=\frac13

. Find the rebound speed and angle.

Section 5 of 14

Explore the graph

Rebound angle against e

Loading the interactive graph…

Try this

$x$ is $e$ ; $y$ is the rebound angle to the wall in degrees.
At $e=1$ the rebound angle equals the impact angle.

$\tan\beta=e\tan\alpha$ .

In the real worldAir hockey. The puck floats on air, so it's nearly smooth: its bounces off the rails follow

\tan\beta=e\tan\alpha

closely.

Section 6 of 14

Common mistakes

IncorrectApplying

e

to the whole speed

CorrectOnly the normal component

The parallel component is unchanged for smooth surfaces.

IncorrectAngle measured from the normal

CorrectCheck which angle is given

To the wall vs to the normal swaps sin and cos.

IncorrectMomentum perpendicular to line of centres changes

CorrectEach sphere keeps it

No impulse in that direction.

Parallel component during impact

Nothing happened. I'm smooth.

Section 7 of 14

Exam tip

Exam tipVector form. If velocities are given as vectors, find the unit vector along the line of centres and use dot products for the components.

Section 8 of 14

Guided practice

SA-FM-C2-G011 markL1 FoundationCalculatorOblique wall

A smooth ball hits a smooth wall at

10

m s

^{-1}

60^\circ

to the wall. Find the velocity component parallel to the wall after impact.

Salam Academy Original Practice · Proposed mark scheme

Section 9 of 14

Practice questions

SA-FM-C2-0012 marksL2 StandardCalculatorOblique wall

With

e=\frac13

in the last question, find the angle (degrees) between the rebound direction and the wall.

Salam Academy Original Practice · Proposed mark scheme

SA-FM-C2-0022 marksL2 StandardCalculatorOblique wall

Find the rebound speed in the same situation (3 s.f.).

Salam Academy Original Practice · Proposed mark scheme

SA-FM-C2-0032 marksL3 ExamNon-calculatorTwo spheres

Smooth sphere A moving at

(3\mathbf i+4\mathbf j)

m s

^{-1}

hits an identical sphere B at rest. The line of centres is parallel to

\mathbf i

and

e=1

. Find the

\mathbf j

-component of A's velocity 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-C2-0043 marksL4 ChallengingNon-calculatorTwo spheres

In the last question, find the

\mathbf i

-component of B's velocity afterwards.

Salam Academy Original Practice · Proposed mark scheme

Section 11 of 14

Summary

Parallel component kept.
Normal component: 1D rules.
$\tan\beta=e\tan\alpha$ .

Section 12 of 14

Quick quiz

Q1 · SA-FM-C2-Q011 markL1 FoundationNon-calculatorOblique wall

When a smooth sphere hits a smooth wall, which velocity component is unchanged?

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 two dimensions
Expected: Oblique impact of a smooth sphere with a fixed surface and of two smooth spheres; components along and perpendicular to the line of centres; vector form.
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

Past paper questions

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