AQA A-Level Further Mathematics Further mechanics: a complete overview of momentum, energy, circular motion and centre of mass

What Further mechanics actually demands

Further mechanics is one of the optional applied areas of AQA A-Level Further Mathematics. It deepens the mechanics of A-Level Mathematics with collisions, restitution, circular motion and centre of mass, and rewards clear modelling: a labelled force diagram, a consistent sign convention, and disciplined working. This guide walks through all five topics, then sets out the exam patterns AQA repeats.

Dimensional analysis

Dimensional analysis expresses every quantity in terms of mass $M$, length $L$ and time $T$: velocity is $LT^{-1}$, force is $MLT^{-2}$, energy is $ML^2T^{-2}$. You use it to check an equation is consistent (every term has the same dimensions) and to predict the form of a relationship up to a dimensionless constant. It is the safety net that catches algebraic slips across the whole module.

Momentum and collisions

Momentum and collisions applies conservation of linear momentum, $m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2$, with impulse as the change in momentum $J = mv - mu$. The coefficient of restitution $e$ links the speeds through Newton's experimental law, $e = \frac{\text{speed of separation}}{\text{speed of approach}}$, with $0 \leq e \leq 1$. Oblique impacts with a wall keep the parallel component and reverse and scale the perpendicular one; successive bounces give geometric series.

Work, energy and power

Work, energy and power uses $W = Fd\cos\theta$, kinetic energy $\frac{1}{2}mv^2$ and potential energy $mgh$. The work-energy principle sets total work equal to the change in kinetic energy, which handles variable forces and resistance neatly. When only gravity acts, mechanical energy is conserved. Power is the rate of doing work, with $P = Fv$ for a driving force.

Circular motion

Circular motion links linear and angular speed by $v = r\omega$, with centripetal acceleration $\frac{v^2}{r} = r\omega^2$ directed to the centre and a required centripetal force $\frac{mv^2}{r}$. Horizontal problems include the conical pendulum and banked tracks; in a vertical circle the speed varies, so you combine energy conservation with the radial equation of motion to find tension or the minimum speed at the top.

Centre of mass

Centre of mass is the mass-weighted average position, $\bar{x} = \frac{\sum m_i x_i}{\sum m_i}$. For composite bodies you treat each part as a particle at its own centre, using negative mass for a removed region. A suspended body hangs with its centre of mass directly below the pivot, and a body on a slope topples when the vertical through its centre of mass leaves the base.

How Further mechanics is examined

A typical AQA profile for Further mechanics:

• Standard model questions. A single collision with restitution, a work-energy calculation with resistance, or a conical pendulum.
• Multi-step problems. Successive collisions, a full vertical-circle analysis, or a composite centre of mass leading to a toppling or suspension condition.
• Reasoning and checking. Dimensional consistency of a formula and predicting the form of a relationship.

Check your knowledge

A mix of recall and calculation questions across Further mechanics. Attempt them under timed conditions, then check against the solutions.

1. Write the dimensions of energy in terms of $M$, $L$ and $T$. (2 marks)
2. A $3$ kg mass changes velocity from $2$ m/s to $6$ m/s. Find the impulse. (2 marks)
3. Two particles approach at $10$ m/s and separate at $4$ m/s. Find $e$. (2 marks)
4. A force of $40$ N moves a body $5$ m in the direction of the force. Find the work done. (2 marks)
5. A particle moves in a circle of radius $2$ m at $3$ rad/s. Find its linear speed. (2 marks)
6. Find the centripetal force on a $5$ kg mass moving at $4$ m/s in a circle of radius $2$ m. (2 marks)
7. Masses $2$ kg and $6$ kg sit at $x = 0$ and $x = 8$. Find $\bar{x}$. (2 marks)
8. State the condition for a suspended lamina to be in equilibrium. (2 marks)