Forces and Newton's laws of motion

What you'll learn

Forces are pushes or pulls that change the motion or shape of objects. This topic covers how forces affect motion through Newton's three laws, how to calculate resultant forces, and the relationship between mass, force and acceleration. You'll learn to apply these principles to everyday situations and solve numerical problems that frequently appear in CIE IGCSE examinations.

Key terms and definitions

Force — a push or pull acting on an object, measured in newtons (N), that can change an object's velocity or shape.

Resultant force — the single force that has the same effect as all the individual forces acting on an object combined.

Mass — the quantity of matter in an object, measured in kilograms (kg); a measure of how difficult it is to change an object's motion.

Weight — the force acting on an object due to gravity, measured in newtons (N).

Acceleration — the rate of change of velocity, measured in metres per second squared (m/s²).

Inertia — the tendency of an object to resist changes in its state of motion.

Newton — the SI unit of force; one newton is the force needed to give a 1 kg mass an acceleration of 1 m/s².

Friction — a force that opposes motion between two surfaces in contact.

Core concepts

Forces as vectors

Forces are vector quantities, meaning they have both magnitude and direction. When multiple forces act on an object, you must consider their directions to find the resultant force.

Representing forces:

• Forces are shown as arrows in diagrams
• The length of the arrow represents the magnitude
• The direction of the arrow shows the direction of the force
• Forces in opposite directions are subtracted
• Forces in the same direction are added

Balanced and unbalanced forces:

• When forces are balanced, the resultant force is zero
• The object remains stationary or continues moving at constant velocity
• When forces are unbalanced, there is a non-zero resultant force
• The object accelerates in the direction of the resultant force

Weight and mass

Weight and mass are fundamentally different, though students often confuse them.

Key differences:

• Mass is a scalar quantity measured in kilograms
• Weight is a force (vector quantity) measured in newtons
• Mass remains constant everywhere in the universe
• Weight varies depending on gravitational field strength

Calculating weight:

The relationship between weight, mass and gravitational field strength is:

W = m × g

Where:

• W = weight (N)
• m = mass (kg)
• g = gravitational field strength (N/kg)

On Earth, g ≈ 10 N/kg (sometimes given as 9.8 N/kg or 9.81 N/kg depending on the question).

Examples:

• A 5 kg object on Earth: W = 5 × 10 = 50 N
• A 60 kg person on Earth: W = 60 × 10 = 600 N
• The same person on the Moon (g ≈ 1.6 N/kg): W = 60 × 1.6 = 96 N

Note that the person's mass remains 60 kg on both Earth and the Moon.

Newton's First Law of Motion

Statement: An object will remain at rest or continue to move at constant velocity unless acted upon by a resultant force.

This law describes inertia — the tendency of objects to maintain their state of motion.

Applications:

• A book resting on a table stays at rest because the forces are balanced (weight downwards, normal reaction upwards)
• A passenger in a car lurches forward when the car brakes suddenly because their body tends to continue at constant velocity
• A spacecraft in deep space continues moving at constant velocity with no fuel needed because there's no friction

Common misconception: Students often think that objects need a force to keep moving. This is incorrect. Objects only need a resultant force to change their motion (to accelerate, decelerate or change direction). Constant velocity requires zero resultant force.

Newton's Second Law of Motion

Statement: The acceleration of an object is directly proportional to the resultant force acting on it and inversely proportional to its mass.

This is expressed mathematically as:

F = m × a

Where:

• F = resultant force (N)
• m = mass (kg)
• a = acceleration (m/s²)

Rearranging the equation:

• To find acceleration: a = F ÷ m
• To find mass: m = F ÷ a

Key implications:

• Greater force → greater acceleration (if mass stays constant)
• Greater mass → smaller acceleration (if force stays constant)
• Doubling the force doubles the acceleration
• Doubling the mass halves the acceleration

Practical examples:

• A car with a more powerful engine can accelerate faster
• A heavy lorry requires more force to achieve the same acceleration as a small car
• Pushing an empty shopping trolley produces greater acceleration than pushing a full one with the same force

Newton's Third Law of Motion

Statement: When object A exerts a force on object B, object B exerts an equal and opposite force on object A.

These are sometimes called action and reaction forces.

Important characteristics:

• The forces are equal in magnitude
• The forces are opposite in direction
• The forces act on different objects
• The forces are the same type (e.g., both gravitational, both contact forces)

Examples:

• When you sit on a chair, you push down on the chair with your weight; the chair pushes up on you with an equal force
• A rocket expels hot gases downwards (action); the gases push the rocket upwards (reaction)
• The Earth pulls you down with gravitational force; you pull the Earth up with an equal gravitational force
• When a swimmer pushes backwards on the water, the water pushes the swimmer forwards

Common mistake: Students sometimes identify balanced forces on the same object as Newton's third law pairs. Remember: third law pairs always act on different objects.

Friction and air resistance

Friction is a force that opposes motion between surfaces in contact. Air resistance (drag) is a type of friction that opposes motion through air.

Characteristics of friction:

• Always acts in the opposite direction to motion
• Converts kinetic energy to thermal energy
• Depends on the surfaces in contact and the normal force between them
• Can be reduced by lubrication or using smoother surfaces

Terminal velocity:

When an object falls through air:

1. Initially, weight > air resistance, so it accelerates downwards
2. As velocity increases, air resistance increases
3. Eventually, air resistance = weight (balanced forces)
4. The object stops accelerating and falls at constant velocity
5. This constant velocity is called terminal velocity

Skydivers experience this phenomenon. Opening a parachute suddenly increases air resistance, creating an upward resultant force that decelerates the diver until a new, lower terminal velocity is reached.

Calculating resultant forces

To find resultant forces, consider all forces acting on an object and their directions.

For forces along a straight line:

• Choose a positive direction
• Add forces in the positive direction
• Subtract forces in the negative direction
• The result is the resultant force

Example: A car experiences:

• Forward driving force: 2000 N
• Backward friction force: 500 N
• Resultant force = 2000 - 500 = 1500 N forwards

Worked examples

Example 1: Weight calculation

Question: A shopping bag has a mass of 3.5 kg. Calculate its weight on Earth. (g = 10 N/kg)

Solution:

• Use the equation: W = m × g
• W = 3.5 × 10
• W = 35 N

Answer: 35 N (accept 35 newtons)

Mark scheme note: 1 mark for correct substitution, 1 mark for correct answer with unit.

Example 2: Newton's second law

Question: A resultant force of 600 N acts on a car of mass 1200 kg.

(a) Calculate the acceleration of the car. [2]

(b) Explain what happens to the acceleration if the mass of the car doubles but the force remains constant. [2]

Solution:

(a)

• Use F = m × a, rearranged to a = F ÷ m
• a = 600 ÷ 1200
• a = 0.5 m/s²

(b)

• If mass doubles, acceleration halves (1 mark)
• The new acceleration would be 0.25 m/s² (1 mark)

Alternative for (b): "Acceleration is inversely proportional to mass, so doubling mass halves acceleration" would receive both marks.

Mark scheme note: For (a), 1 mark for correct rearrangement/substitution, 1 mark for answer with unit. For (b), 1 mark for explaining inverse relationship, 1 mark for correct application.

Example 3: Terminal velocity

Question: A skydiver jumps from an aircraft.

(a) Explain why the skydiver initially accelerates. [2]

(b) Explain why the skydiver eventually falls at constant velocity. [2]

(c) State the name given to this constant velocity. [1]

Solution:

(a)

• The weight force is greater than the air resistance (1 mark)
• This produces a resultant force downwards / unbalanced forces (1 mark)

(b)

• As velocity increases, air resistance increases (1 mark)
• Eventually air resistance equals weight, so resultant force becomes zero / forces become balanced (1 mark)

(c) Terminal velocity

Mark scheme note: Accept "drag" instead of "air resistance". For (a), both parts must be present for full marks. For (b), must mention that forces become equal/balanced.

Common mistakes and how to avoid them

• Confusing mass and weight: Remember that mass is measured in kg and is constant everywhere; weight is measured in N and varies with gravitational field strength. Always use W = m × g to convert between them.

• Forgetting units in calculations: Every numerical answer needs a unit. Force is measured in newtons (N), mass in kilograms (kg), acceleration in m/s², and weight in newtons (N). The mark scheme often reserves one mark specifically for the correct unit.

• Misidentifying Newton's third law pairs: Third law pairs must act on different objects and be the same type of force. The forces holding a book at rest on a table (weight and normal reaction) are balanced forces, not a third law pair.

• Thinking objects need force to maintain motion: According to Newton's first law, objects need zero resultant force to maintain constant velocity. Force is only needed to change motion (accelerate).

• Not showing working in calculations: Even if you get the correct answer, you may lose marks without showing your method. Always write the equation, substitute values, and show your calculation steps.

• Confusing acceleration and velocity: An object at terminal velocity has zero acceleration (not zero velocity). When resultant force is zero, acceleration is zero, but velocity can be any constant value.

Exam technique for "Forces and Newton's laws of motion"

• Command word awareness: "State" requires a brief answer without explanation (1 mark). "Explain" requires a reason or mechanism (usually 2+ marks). "Calculate" requires you to show numerical working with the correct formula, substitution, and answer with units.

• Drawing force diagrams: Use a ruler for arrows, label each force clearly, and ensure arrow lengths approximately represent force magnitudes. In equilibrium situations, show opposing forces as equal length.

• Multi-step calculations: For questions worth 3-4 marks, you'll likely need to use two equations or perform two calculation steps. Identify what you're given, what you need to find, and which equations link them.

• Extended response questions: When explaining applications of Newton's laws (typically worth 4-6 marks), structure your answer logically: describe the situation, identify the forces, state which law applies, and explain the consequence for motion. Use scientific terminology consistently.

Quick revision summary

Forces are measured in newtons and can change motion or shape. Weight (W = m × g) differs from mass. Newton's first law: objects maintain constant velocity unless a resultant force acts. Newton's second law: F = m × a relates force, mass and acceleration. Newton's third law: action and reaction forces are equal, opposite and act on different objects. Terminal velocity occurs when drag equals weight. Always include units, show working, and distinguish between force types in exam answers.