What is in the Edexcel Further Mechanics optional papers?

Further Mechanics extends mechanics beyond A-Level Mathematics. It covers momentum and impulse in one and two dimensions, work energy and power, elastic collisions using Newton's law of restitution, and circular motion in horizontal and vertical circles, alongside elastic strings and springs and further dynamics. A student takes it as one or both of their two optional 9FM0 papers.

How does Further Mechanics fit into 9FM0?

Edexcel A-Level Further Mathematics has four equally weighted papers. Two are compulsory Core Pure; the other two are chosen from Further Pure, Further Statistics, Further Mechanics and Decision Mathematics. A student selects two optional papers, so Further Mechanics can be one or both of them.

Which Further Mechanics topics do students find hardest?

Oblique impact of smooth spheres, combining the law of restitution with conservation of momentum, the conical pendulum, and motion in a vertical circle where speed varies and you combine energy conservation with the radial equation of motion. Clear diagrams and resolved components are essential.

How should I revise Further Mechanics?

Drill the core results until automatic: impulse equals change in momentum, the work-energy principle, Newton's law of restitution, and the centripetal force equation. Draw a clear force diagram for every problem, resolve into sensible directions, and write each equation of motion explicitly.

Do I need A-Level Mechanics for Further Mechanics?

Yes. Further Mechanics assumes the mechanics within A-Level Mathematics, including forces, Newton's laws and kinematics. It also draws on Core Pure calculus, so secure both before tackling the optional Further Mechanics content.

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Edexcel A-Level Further Mathematics Further Mechanics: momentum, energy, collisions and circular motion

A deep-dive Edexcel A-Level Further Mathematics guide to the optional Further Mechanics papers. Covers momentum and impulse, work energy and power, elastic collisions and the law of restitution, and circular motion in horizontal and vertical circles, with the modelling and exam patterns Edexcel rewards.

Generated by Claude Opus 4.818 min read9FM0Updated 2026-06-02

Reviewed by: AI editorial process; not yet individually human-reviewed

Jump to a section

What Further Mechanics demands
Momentum and impulse
Work, energy and power
Elastic collisions
Circular motion
How the Further Mechanics papers are examined
Check your knowledge

What Further Mechanics demands

The Further Mechanics options extend mechanics beyond A-Level Mathematics and are taken as one or both of the two optional 9FM0 papers. They reward clear modelling: a labelled force diagram, sensibly resolved components, and explicit equations of motion. This guide walks through the four headline topics and the exam patterns Edexcel repeats. Each topic has a matching dot-point page with worked questions.

Momentum and impulse

Momentum and impulse treats momentum $\mathbf{p} = m\mathbf{v}$ , impulse as the change in momentum (and the area under a force-time graph), the impulse-momentum principle, and conservation of momentum in collisions and explosions, in one and two dimensions by resolving into components.

Work, energy and power

Work, energy and power covers work done by a force $W = Fd\cos\theta$ , kinetic energy $\frac{1}{2}mv^2$ and potential energy $mgh$ , the work-energy principle, conservation of mechanical energy where only conservative forces act, and power as the rate of working $P = Fv$ .

Elastic collisions

Elastic collisions applies Newton's experimental law of restitution, separation speed equals $e$ times approach speed, paired with conservation of momentum for direct impacts, extends to oblique impact of smooth spheres and impact with a fixed surface, and finds the kinetic energy lost (zero only when $e = 1$ ).

Circular motion

Circular motion uses $v = r\omega$ and acceleration $\frac{v^2}{r} = r\omega^2$ towards the centre, analyses horizontal circles including the conical pendulum, and treats motion in a vertical circle by combining conservation of energy for speed with the radial equation of motion for the tension or reaction.

How the Further Mechanics papers are examined

A typical Edexcel profile:

Short calculation questions. An impulse, a kinetic energy, a power, or a centripetal force.
Multi-step modelling. A two-sphere collision with restitution, a conical pendulum, or a full vertical-circle problem.
Two-dimensional work. Oblique impact and momentum conserved in perpendicular directions.
Reasoning in context. Justifying assumptions such as smooth surfaces or light strings.

Check your knowledge

Attempt these under timed conditions, then check the solutions.

A force of $6\,\text{N}$ acts for $3\,\text{s}$ . Find the impulse. (1 mark)
State the impulse-momentum principle in symbols. (1 mark)
Find the kinetic energy of a $4\,\text{kg}$ mass moving at $5\,\text{m s}^{-1}$ . (2 marks)
A car works at $9\,000\,\text{W}$ at $15\,\text{m s}^{-1}$ . Find the driving force. (2 marks)
State Newton's law of restitution. (1 mark)
Give the range of the coefficient of restitution. (1 mark)
Write the centripetal acceleration in terms of $r$ and $\omega$ . (1 mark)
Find the minimum speed at the top of a vertical circle of radius $1\,\text{m}$ for the string to stay taut (use $g = 9.8$ ). (2 marks)

Solutions

Eight short questions covering the Further Mechanics topics.

Step 1: Q1 - Impulse from a constant force

Impulse equals force multiplied by the time it acts. This is the simplest application of the impulse formula, with both quantities given directly.

J = Ft = 6 \times 3 = 18\,\text{N s}

Final answer: $18\,\text{N s}$ .

Step 2: Q2 - The impulse-momentum principle in symbols

The impulse-momentum principle states that the impulse acting on a body equals the change in its momentum. The vector form makes clear that direction matters in every collision problem.

\mathbf{J} = m\mathbf{v} - m\mathbf{u}

Final answer: $\mathbf{J} = m\mathbf{v} - m\mathbf{u}$ .

Step 3: Q3 - Kinetic energy of a moving mass

Kinetic energy depends on both the mass and the square of the speed. Substituting $m = 4\,\text{kg}$ and $v = 5\,\text{m s}^{-1}$ into the standard formula gives the result directly.

\operatorname{KE} = \frac{1}{2}(4)(5^2) = 50\,\text{J}

Final answer: $50\,\text{J}$ .

Step 4: Q4 - Driving force from power and speed

Power is the product of force and speed, so rearranging $P = Fv$ gives $F = P/v$ . Both quantities are given, so a single substitution yields the force.

F = \frac{P}{v} = \frac{9000}{15} = 600\,\text{N}

Final answer: $600\,\text{N}$ .

Step 5: Q5 - Newton's law of restitution

Newton's experimental law relates the speeds of separation and approach along the line of impact. The coefficient of restitution $e$ is the ratio of these speeds.

Speed of separation $= e \times$ speed of approach.

Final answer: Speed of separation equals $e$ times the speed of approach.

Step 6: Q6 - Range of the coefficient of restitution

A perfectly elastic collision has $e = 1$ and no kinetic energy is lost. A perfectly inelastic collision has $e = 0$ and the objects coalesce. All real collisions lie strictly between these extremes.

0 \le e \le 1

Final answer: $0 \le e \le 1$ .

Step 7: Q7 - Centripetal acceleration in terms of $r$ and $\omega$

Centripetal acceleration points towards the centre and can be expressed using the angular velocity $\omega$ and the radius $r$ . This form is particularly useful when the angular speed is given rather than the linear speed.

a = r\omega^2

Final answer: $a = r\omega^2$ .

Step 8: Q8 - Minimum speed at the top of a vertical circle

At the top of a vertical circle, the string or track can only pull or push inwards. The minimum speed occurs when the tension or normal reaction is exactly zero, so gravity alone provides the centripetal force: $mg = \frac{mv^2}{r}$ , giving $v^2 = gr$ .

v^2 = gr = 9.8 \times 1 = 9.8 \implies v \approx 3.13\,\text{m s}^{-1}

Final answer: Approximately $3.13\,\text{m s}^{-1}$ .

Sources & how we know this

Pearson Edexcel A-Level Further Mathematics (9FM0) specification — Pearson Edexcel (2017)