Newton's laws of motion

CIE A-Level Mathematics· Mechanics 1: 3.2 Newton's laws of motion· 15 min read

1. Newton's Three Laws of Motion★☆☆☆☆⏱ 3 min

📘 Definition

Newton's Three Laws of Motion

First Law: A body remains at rest or constant velocity unless acted on by a resultant external force. 2. Second Law: For constant mass, resultant force equals $F = m a$ . 3. Third Law: If A exerts a force on B, B exerts an equal, opposite force of the same type on A.

Example:

A book on a table: weight and normal reaction are not a third law pair. The third law pair is Earth pulling the book, and the book pulling Earth.

✓ Quick check

Test your understanding:

A tennis ball is hit by a racket. Which is a Newton's third law pair?
- Racket force on ball, ball force on racket
- Weight of ball, air resistance on ball
- Racket force on ball, weight of ball

2. Newton's Second Law in 1 Dimension★★☆☆☆⏱ 5 min

For straight line motion, resolve all forces along the direction of motion, take the direction of acceleration as positive, then apply $F_{n e t} = ma$ to find unknown acceleration or force.

📐 Worked Example

A 5 kg box is pulled along a smooth horizontal surface by a horizontal rope with tension 20 N. Find the acceleration of the box.

1
No friction for a smooth surface, so only horizontal force is tension:
2
$F_{net} = 20 \, \text{N}$
3
Apply Newton's second law:
4
$20 = 5a \implies a = 4 \, \text{m s}^{-2}$

📐 Worked Example

A 1000 kg car moving at 20 m s⁻¹ brakes to rest over 50 m. Find the constant braking force.

1
First find acceleration with kinematics $v^{2} = u^{2} + 2 a s$ :
2
$0^2 = 20^2 + 2a(50) \implies a = -4 \, \text{m s}^{-2}$
3
Negative acceleration means it opposes motion. Resultant force is the braking force $F$ :
4
$-F = ma = 1000(-4) \implies F = 4000 \, \text{N}$

Exam tip:

Always draw a labelled force diagram before writing your force equation. Stick to your sign convention consistently.

3. Newton's Second Law in 2 Dimensions★★★☆☆⏱ 4 min

For motion on inclined planes, resolve forces into two perpendicular directions: one parallel to acceleration (along the plane) and one perpendicular, where acceleration is zero.

📘 Definition

Weight components on an inclined plane

Angle $θ$ to the horizontal

Weight $m g$ splits into two components: parallel to the plane: $m g sin θ$ , perpendicular to the plane: $m g cos θ$ .

📐 Worked Example

A 4 kg particle slides down a rough 25° incline, with friction coefficient 0.2. Find the acceleration.

1
Resolve perpendicular to the plane (acceleration = 0, so resultant force = 0):
2
$R - 4g\cos 25^\circ = 0 \implies R \approx 35.5 \, \text{N}$
3
Friction $F = μ R = 0.2 \times 35.5 \approx 7.1 N$ , acting up the slope:
4
Resolve parallel, take down slope as positive:
5
$4g\sin 25^\circ - F = 4a$
6
$16.6 - 7.1 = 4a \implies a \approx 2.4 \, \text{m s}^{-2}$

Exam tip:

Check your components: if $θ = 0^{\circ}$ (flat plane), parallel component should be zero. If you get $m g$ , you swapped sin and cos.

4. Common Exam Application: Connected Particles Over a Pulley★★★★☆⏱ 5 min

For light inextensible strings over smooth pulleys, tension is the same on both sides, and both particles have the same magnitude of acceleration. Write separate force equations for each particle, then eliminate tension to solve.

📐 Worked Example

Two particles of mass 2 kg and 3 kg are connected by a light string over a smooth pulley. Find acceleration and tension.

1
Let $a$ = acceleration, $T$ = tension. 3 kg accelerates down, 2 kg accelerates up.
2
Newton's second law for 3 kg (down positive):
3
$3g - T = 3a$
4
Newton's second law for 2 kg (up positive):
5
$T - 2g = 2a$
6
Add equations to eliminate T:
7
$g = 5a \implies a = 1.96 \, \text{m s}^{-2}$
8
Substitute back to find T:
9
$T = 2g + 2a = 23.52 \, \text{N}$

Exam tip:

Adding the individual force equations for connected particles always eliminates tension, saving you time in exams.

5. Common Pitfalls

Wrong move:

Treating action-reaction pairs as acting on the same body and cancelling them

Why:

Newton's third law pairs act on different bodies, so they never cancel for a single body

Correct move:

Only include forces acting on the body you are analysing in your resultant force calculation

Wrong move:

Swapping sine and cosine components of weight on an inclined plane

Why:

For angles measured to the horizontal, the parallel component is $m g sin θ$ , not $m g cos θ$

Correct move:

Test for θ=0° (flat plane): if your parallel component is zero, it is correct, otherwise swap them

Wrong move:

Using mass instead of weight (forgetting to multiply by g) in force equations

Why:

Weight is a force, $F = ma$ requires force in newtons, mass in kg

Correct move:

Always write weight as $m g$ , not just $m$ , when calculating resultant force

Wrong move:

Writing $T - m g = ma$ for a mass accelerating downwards

Why:

Resultant force must point in the direction of acceleration

Correct move:

Write $m g - T = ma$ for a mass accelerating downwards to get the correct resultant force direction

Wrong move:

Using $g = 10 m s^{- 2}$ when not specified by the question

Why:

CIE standard requires $g = 9.8 m s^{- 2}$ , wrong g leads to lost marks

Correct move:

Always use $g = 9.8$ unless the question explicitly tells you to use 10

6. Quick Reference Cheatsheet

Concept	Key Rule/Formula
Newton's First Law	No resultant force → constant velocity
Newton's Second Law	$F_{n e t} = ma$ (constant mass)
Newton's Third Law	Equal opposite forces on different bodies
Weight on incline (θ to horizontal)	Parallel: $m g sin θ$ , Perpendicular: $m g cos θ$
Connected particles over pulley	Add individual equations to eliminate tension
CIE required g value	$g = 9.8 m s^{- 2}$

7. Frequently Asked

What value of $g$ should I use in CIE exams?

CIE requires $g = 9.8 m s^{- 2}$ unless the question explicitly specifies $10 m s^{- 2}$ .

Do I need to memorize Newton's laws word for word?

You are rarely asked to state them, but you must understand them correctly to apply to problems. Core statements are worth learning.

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2022 · M1
Connected particles on inclined plane
2023 · M1
Object on accelerating lift
2021 · M1
Two masses over a fixed pulley

Going deeper

syllabusCIE 9709 Official Syllabus

What's Next

Newton's laws of motion are the foundation of all A-Level mechanics, and mastery of this topic is critical for all further mechanics content. The core principle $F_{n e t} = ma$ is used in every dynamic problem you will solve, from friction to circular motion to work and energy. After this, you can move on to more complex applications of Newton's laws, including problems involving friction, more complex connected particle systems, and circular motion. Practise setting up force diagrams correctly to avoid common mistakes and pick up full marks in exams.