12 min readConcept sheetSection A MCQ (1 mark), fill in the blanks (2 marks, two parts) or passage (5 marks); Section B very short answer (2 marks), short answer numerical (3 marks) or long answer (5 marks)

Momentum and Newton's second law

Momentum is mass × velocity, p = mv. The rate of change of momentum equals the force, which gives F = ma.

Momentump = mv (m = mass in kg, v = velocity in m s⁻¹); SI unit kg m s⁻¹; points the same way as the velocity

Newton's second lawThe rate of change of momentum of a body is directly proportional to the force on it and takes place in the direction of the force

Force equationF = ma (F in N, m in kg, a in m s⁻²); also F = (mv − mu)/t

One newtonThe force that gives a 1 kg mass an acceleration of 1 m s⁻²; 1 N = 1 kg m s⁻²

What momentum is

A slow, loaded truck is hard to stop. So is a fast cricket ball. The truck has a large mass; the ball has a large velocity. So the effect of a moving body depends on both its mass and its velocity. Their product is called momentum: p = mv. Its SI unit is kg m s⁻¹. Momentum has a direction, the same as the velocity.

Derivation: from the second law to F = ma

1
Second law: the rate of change of momentum of a body is directly proportional to the force on it, and takes place in the direction of the force.
2
Let a constant force F change the velocity of a body of mass m from u to v in time t.
3
Initial momentum p₁ = mu. Final momentum p₂ = mv. Change in momentum = mv − mu.
4
Rate of change of momentum = (mv − mu)/t = m(v − u)/t.
5
So F ∝ m(v − u)/t, or F = k m(v − u)/t, where k is a constant.
6
But (v − u)/t is the acceleration a. So F = kma.
7
Choose the unit of force so that 1 unit gives 1 kg an acceleration of 1 m s⁻². Then k = 1.
8★
So, F = ma. This unit of force is the newton (N).

★ key step

Worked example: momentum of a scooter

Problem

A scooter and its rider have a total mass of 150 kg and move at 12 m s⁻¹. Find their momentum.

Working

Given: m = 150 kg, v = 12 m s⁻¹.
Formula: p = mv.
p = 150 kg × 12 m s⁻¹ = 1800 kg m s⁻¹.

Answer

1800 kg m s⁻¹, in the direction of motion

Worked example: force and acceleration

Problem

A force gives a 4 kg trolley an acceleration of 2.5 m s⁻². Find the force. What acceleration would the same force give an 8 kg trolley?

Working

Given: m = 4 kg, a = 2.5 m s⁻².
F = ma = 4 kg × 2.5 m s⁻² = 10 N.
For the 8 kg trolley: a = F/m = 10 N ÷ 8 kg = 1.25 m s⁻².

Answer

The force is 10 N. It gives the 8 kg trolley 1.25 m s⁻², half as much, because the mass is double

Worked example: braking force on an auto-rickshaw

Problem

An auto-rickshaw with passengers has a mass of 400 kg and moves at 54 km h⁻¹. The brakes stop it in 5 s. Find the braking force, taking it as constant.

Working

Convert the speed: 54 km h⁻¹ = 54 × 1000 m ÷ 3600 s = 15 m s⁻¹.
Given: m = 400 kg, u = 15 m s⁻¹, v = 0, t = 5 s.
a = (v − u)/t = (0 − 15 m s⁻¹) ÷ 5 s = −3 m s⁻².
F = ma = 400 kg × (−3 m s⁻²) = −1200 N.
The minus sign shows the force acts against the motion.

Answer

A braking force of 1200 N, opposite to the motion

The second law in daily life

Catching a fast ball: a fielder pulls the hands back. The ball takes more time to stop, so the rate of change of momentum is smaller. The force on the hands is smaller and they do not get hurt.
Falling on a hard floor hurts, because the body stops in a very short time, so the force is large. Sand, a mattress or a heap of hay makes the stop last longer, so the force is smaller.
High jumpers and long jumpers land on foam beds or loose sand for the same reason.
A karate player breaks a slab by hitting very fast. The hand's momentum drops to zero in a tiny time, so a very large force acts on the slab.
A bundle lifted with a jerk can snap its string. The momentum changes in a very short time, so the pull on the string is large.
Airbags and crumple zones in cars make a crash stop last longer, so the force on passengers is smaller.

In real life

A fielder pulling back the hands, a wrestler landing on a soft mat and a car airbag all do the same thing. They make the stop take longer, so the force is smaller.

Exam angle

2 marks: “Define momentum and give its SI unit. A 0.5 kg ball moves at 8 m s⁻¹. Find its momentum.” Momentum is the product of mass and velocity, p = mv; SI unit kg m s⁻¹. p = 0.5 kg × 8 m s⁻¹ = 4 kg m s⁻¹.

2 marks: “Why do long jump athletes land in a pit of loose sand and not on a hard floor?” Sand lets the athlete stop over a longer time. So the rate of change of momentum is smaller, and by the second law the force on the body is smaller. This prevents injury.

3 marks: “A force acts for 4 s on a 2000 kg car at rest and gives it a speed of 10 m s⁻¹. Find the acceleration and the force.” Method: a = (v − u)/t, then F = ma. a = (10 − 0) ÷ 4 = 2.5 m s⁻². F = 2000 × 2.5 = 5000 N. Answer: 2.5 m s⁻² and 5000 N.

Remember it

Force is how fast momentum changes: stop slowly and you feel less force.