Further Mathematics · Lesson 20

Momentum, impulse, work–energy & friction (mechanics)

WAECNECO

What the syllabus asks

WAEC: linear momentum and impulse; conservation of linear momentum; direct impact of two bodies, coalescence, recoil; Newton’s law of restitution (introduction); work, energy (kinetic and potential), power; work–energy principle; friction: limiting friction, coefficient of friction, angle of friction, motion on rough horizontal and inclined planes

NECO: linear momentum and impulse; conservation of linear momentum; direct impact of two bodies, coalescence, recoil; Newton’s law of restitution (introduction); work, energy (kinetic and potential), power; work–energy principle; friction: limiting friction, coefficient of friction, angle of friction, motion on rough horizontal and inclined planes

What you will learn

Calculate momentum and impulse, and use Ft = mv − mu.
Apply conservation of momentum to collisions, coalescence, explosions and recoil.
Use Newton’s law of restitution for direct impact of two spheres.
Calculate work, kinetic energy, potential energy and power.
Apply the work–energy principle, including loss of energy in collisions.
Use F ≤ μR to solve problems on rough horizontal and inclined planes.
Relate the angle of friction to the coefficient of friction.

Key points

Momentum p = mv (vector), unit kg m/s = N s.
Impulse Ft = mv − mu; longer contact time means smaller force.
Conservation: m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂; coalescence uses (m₁ + m₂)v.
Explosions and recoil: total momentum stays zero.
Restitution: v₂ − v₁ = e(u₁ − u₂); e = 1 elastic, e = 0 coalesce.
Work W = Fs cos θ; no work by forces perpendicular to motion.
KE = ½mv², PE = mgh; net work = ΔKE; mechanical energy conserved without friction.
Power P = W/t = Fv; at maximum speed, driving force = resistance.
Friction F ≤ μR; R = mg on level ground, mg cos θ on an incline.
Angle of friction: tan λ = μ; a block slips when tan θ exceeds μ.

Start here: Linear momentum

A slow lorry and a fast football can both knock you over. A heavy lorry moving slowly is hard to stop; so is a light bullet moving very fast. The quantity that measures this "quantity of motion" is momentum.

Linear momentum p = mass × velocity = mv. Unit: kg m/s (equivalently N s). Momentum is a vector: it has the direction of the velocity.

Because momentum is a vector, choose a positive direction and give motion the other way a negative sign.

Worked example 20.1

Find the momentum of (a) a 2000 kg car at 15 m/s, (b) a 0.02 kg bullet at 400 m/s, (c) a 0.5 kg ball moving at 6 m/s to the left (take right as positive).

(a) p = 2000 × 15 = 30,000 kg m/s.
(b) p = 0.02 × 400 = 8 kg m/s.
(c) p = 0.5 × (−6) = −3 kg m/s (3 kg m/s to the left).

30,000; 8; −3 kg m/s

Worked example 20.2 (change in momentum)

A 0.4 kg ball moving at 20 m/s hits a wall and rebounds at 15 m/s. Find the change in momentum.

Take the original direction as positive: u = +20, v = −15.
Δp = m(v − u) = 0.4(−15 − 20) = 0.4 × (−35) = −14 kg m/s.
The magnitude is 14 kg m/s, directed away from the wall.

14 N s away from the wall

⚠️ Common mistake: subtracting speeds (20 − 15 = 5) when a body rebounds. The velocity changes sign, so the speeds add: 20 + 15 = 35.

Try it yourself. A 60 kg athlete runs at 9 m/s. Find his momentum.

Show solution

60 × 9 = 540 kg m/s.

Continue this lesson in the free app →

Practice questions

1. Find the momentum of a 1500 kg bus moving at 12 m/s.

Show worked answer

18,000 kg m/s.

2. A 0.3 kg ball at 10 m/s is stopped in 0.02 s. Find the average force.

Show worked answer

3/0.02 = 150 N.

3. A 6 kg body at 5 m/s collides with a 4 kg body at rest; they move together. Find the common velocity.

Show worked answer

30 = 10v → 3 m/s.

4. A 3 kg gun fires a 0.015 kg bullet at 500 m/s. Find the recoil speed.

Show worked answer

7.5/3 = 2.5 m/s.

5. Find the work done when a 40 N force at 60° to the horizontal moves a box 10 m horizontally.

Show worked answer

40 × 10 × 0.5 = 200 J.

Key words

Momentum: Mass × velocity; a vector.
Impulse: Force × time; equals change in momentum.
Conservation of momentum: Total momentum of a system is constant if no external force acts.
Coalesce: Two bodies join and move together after impact.
Recoil: The backward motion of a gun or body when it fires or throws something forward.
Coefficient of restitution e: Speed of separation ÷ speed of approach in a direct impact; 0 ≤ e ≤ 1.
Work: Force × distance moved in the direction of the force; unit joule.
Kinetic energy: Energy of motion, ½mv².
Potential energy: Energy due to position, mgh.
Power: Rate of doing work; unit watt (J/s).
Friction: A force opposing relative motion between surfaces in contact.
Limiting friction: The maximum frictional force, μR, when sliding is about to start.

Learn Momentum, impulse, work–energy & friction (mechanics) step by step, free.
The Cara's Tips app teaches every topic from the very basics, with quick checks, worked examples and exam practice. It works offline.
Open this lesson in the free app Download for Android