Newton's Second Law

What you'll learn

What resultant force means, and why it matters for motion.
How force, mass and acceleration are linked.
How to use and rearrange $F = ma$ .
How the required practical tests Newton’s Second Law.

Start point: forces and resultant force

A force is a push or pull. Force is measured in newtons, N.

A force is a vector, which means it has both a size and a direction. So when forces act in opposite directions, you must think about direction, not just add every number together.

Definition

Resultant force

The resultant force is the single overall force left after all the forces on an object have been combined, taking their directions into account.

If the resultant force is zero, the forces are balanced. The object either stays still or continues moving at constant velocity. Velocity means speed in a particular direction.

If the resultant force is not zero, the forces are unbalanced, and the object accelerates in the direction of the resultant force.

The diagram shows a trolley with forces acting on it. The upward normal contact force is the support force from the track. Weight is the downward force due to gravity. Friction and drag are forces that oppose motion.

Force diagram for a trolley showing resultant force causing acceleration

Example

Finding a resultant force

Take right as the positive direction. A 90 N pull acts to the right and 25 N friction acts to the left, so $\text{ N} - 25 \text{ N}$ .
Calculate the resultant force: $\text{ N}$ .
The answer is positive, so the resultant force is 65 N to the right. Any acceleration will also be to the right.

Acceleration: changing velocity

Acceleration is the rate of change of velocity. It tells you how quickly velocity changes each second.

Acceleration is measured in metres per second squared, m/s^2, and its symbol is $a$ .

You may already know:

\frac{v-u}{t}

where $u$ is the initial velocity, $v$ is the final velocity, and $t$ is the time taken.

Example

Calculating acceleration from a velocity change

A cyclist speeds up from 2.0 m/s to 8.0 m/s in 3.0 s, so $\text{ m/s}$ , $\text{ m/s}$ and $\text{ s}$ .
Substitute into the acceleration equation: $\frac{8.0 \text{ m/s} - 2.0 \text{ m/s}}{3.0 \text{ s}}$ .
Calculate $\text{ m/s}^2$ , so the cyclist’s velocity increases by 2.0 m/s every second.

Newton’s Second Law in words

Newton’s Second Law links the acceleration of an object to two things:

the resultant force acting on it
the mass of the object

Mass is the amount of matter in an object. It is measured in kilograms, kg. In this topic, mass tells you how difficult it is to change an object’s motion.

Definition

Proportional

Two quantities are proportional if changing one by a scale factor changes the other by the same scale factor. The symbol for “is proportional to” is $∝\propto$ .

For a constant mass:

\propto F

So if the resultant force doubles, the acceleration doubles.

For a constant resultant force:

\propto \frac{1}{m}

This means acceleration is inversely proportional to mass. If the mass doubles, the acceleration halves.

Key Idea

Newton’s Second Law in words

The acceleration of an object is proportional to the resultant force acting on it, and inversely proportional to the mass of the object.

Newton’s Second Law as an equation

The equation you need to recall and apply is:

F = ma

where:

$F$ is the resultant force in newtons, N
$m$ is the mass in kilograms, kg
$a$ is the acceleration in metres per second squared, m/s^2

Tip

Rearranging F = ma

If acceleration is unknown, use $\frac{F}{m}$ . If mass is unknown, use $\frac{F}{a}$ .

Example

Using F = ma with resistive forces

A car has a driving force of 2500 N forwards and resistive forces of 700 N backwards, so the resultant force is $\text{ N} - 700 \text{ N} = 1800 \text{ N}$ forwards.
The car’s mass is 1000 kg and acceleration is unknown, so use $\frac{F}{m}$ .
Substitute: $\frac{1800 \text{ N}}{1000 \text{ kg}} = 1.8 \text{ m/s}^2$ . The car accelerates forwards at 1.8 m/s^2.

Common Mistake

Using the wrong force

Do not put the engine thrust, pull or push into $F = ma$ unless it is already the resultant force. If friction, drag or another opposing force is given, combine the forces first.

Higher Tier: inertial mass

This part is Higher Tier only.

Definition

Inertial mass

Inertial mass is a measure of how difficult it is to change the velocity of an object. It is defined as the ratio of force over acceleration: $\frac{F}{a}$ .

A larger inertial mass means the same resultant force produces a smaller acceleration. This is why a loaded van is harder to get moving than an empty van.

Estimating forces in road transport

In exams, you may be asked to estimate speeds, accelerations and forces for everyday transport.

The symbol $∼\sim$ means “approximately” or “of the order of”. For example, a typical car might have $\sim 1000 \text{ kg}$ .

Useful rough values:

a cyclist and bike: about 100 kg
a small car: about 1000 kg to 1500 kg
motorway speed: about 30 m/s
a large car acceleration or braking acceleration: a few m/s^2

Example

Estimating a braking force

A 1200 kg car slows from 20 m/s to 0 m/s in 4.0 s, so $\frac{0 \text{ m/s} - 20 \text{ m/s}}{4.0 \text{ s}} = -5.0 \text{ m/s}^2$ .
Use Newton’s Second Law: $\text{ kg} \times -5.0 \text{ m/s}^2 = -6000 \text{ N}$ .
The negative sign means the force acts opposite to the motion. As an estimate, the braking force is $\sim 6000 \text{ N}$ backwards.

Required practical activity 7: investigating Newton’s Second Law

You need to know how to investigate:

the effect of varying the force on the acceleration of an object of constant mass
the effect of varying the mass on the acceleration produced by a constant force

A common setup uses a dynamics trolley on a track, a string over a pulley, and a hanging mass. Light gates are sensors that time a card as it passes through them. A data logger records the timings and can calculate acceleration.

Required practical setup with trolley, pulley, hanging masses and light gates

Varying force while keeping mass constant

To test how force affects acceleration:

Set up the trolley, string, pulley and hanging mass.
Keep the total moving mass constant.
Increase the driving force by moving slotted masses from the trolley to the hanging holder.
Release the trolley without pushing it.
Measure the acceleration using light gates or a data logger.
Repeat each reading and calculate a mean.

You should find that acceleration increases in direct proportion to resultant force.

Varying mass while keeping force constant

To test how mass affects acceleration:

Keep the hanging mass the same, so the driving force stays constant.
Add masses to the trolley to increase the total moving mass.
Measure the acceleration each time.
Repeat readings and calculate a mean.

You should find that increasing mass decreases acceleration.

The expected graph shapes are shown below.

Graphs showing acceleration against force, mass, and reciprocal mass

Common Mistake

Changing two variables at once

When testing the effect of force, keep the total moving mass constant by moving masses between the trolley and the hanging holder. Do not simply add extra masses to the hanger.

Exam technique

In the exam

Find the resultant force before using $F = ma$ .
Check units: force in N, mass in kg, acceleration in m/s^2.
For estimates, use rounded values and the symbol $∼\sim$ when your answer is approximate.
In practical questions, state what is changed, what is kept constant, and how acceleration is measured.

Self review

Check yourself

If a 2 kg object has a resultant force of 10 N, what is its acceleration?
Why does doubling the mass halve the acceleration when the resultant force stays the same?
In the trolley practical, how can you vary the force while keeping the total mass constant?

Newton's Second Law

What you'll learn

What resultant force means, and why it matters for motion.
How force, mass and acceleration are linked.
How to use and rearrange $F = ma$ .
How the required practical tests Newton’s Second Law.

Start point: forces and resultant force

A force is a push or pull. Force is measured in newtons, N.

A force is a vector, which means it has both a size and a direction. So when forces act in opposite directions, you must think about direction, not just add every number together.

Definition

Resultant force

The resultant force is the single overall force left after all the forces on an object have been combined, taking their directions into account.

If the resultant force is zero, the forces are balanced. The object either stays still or continues moving at constant velocity. Velocity means speed in a particular direction.

If the resultant force is not zero, the forces are unbalanced, and the object accelerates in the direction of the resultant force.

Force diagram for a trolley showing resultant force causing acceleration

Example

Finding a resultant force

Take right as the positive direction. A 90 N pull acts to the right and 25 N friction acts to the left, so $\text{ N} - 25 \text{ N}$ .
Calculate the resultant force: $\text{ N}$ .
The answer is positive, so the resultant force is 65 N to the right. Any acceleration will also be to the right.

Acceleration: changing velocity

Acceleration is the rate of change of velocity. It tells you how quickly velocity changes each second.

Acceleration is measured in metres per second squared, m/s^2, and its symbol is $a$ .

You may already know:

\frac{v-u}{t}

where $u$ is the initial velocity, $v$ is the final velocity, and $t$ is the time taken.

Example

Calculating acceleration from a velocity change

A cyclist speeds up from 2.0 m/s to 8.0 m/s in 3.0 s, so $\text{ m/s}$ , $\text{ m/s}$ and $\text{ s}$ .
Substitute into the acceleration equation: $\frac{8.0 \text{ m/s} - 2.0 \text{ m/s}}{3.0 \text{ s}}$ .
Calculate $\text{ m/s}^2$ , so the cyclist’s velocity increases by 2.0 m/s every second.

Newton’s Second Law in words

Newton’s Second Law links the acceleration of an object to two things:

the resultant force acting on it
the mass of the object

Mass is the amount of matter in an object. It is measured in kilograms, kg. In this topic, mass tells you how difficult it is to change an object’s motion.

Definition

Proportional

Two quantities are proportional if changing one by a scale factor changes the other by the same scale factor. The symbol for “is proportional to” is $∝\propto$ .

For a constant mass:

\propto F

So if the resultant force doubles, the acceleration doubles.

For a constant resultant force:

\propto \frac{1}{m}

This means acceleration is inversely proportional to mass. If the mass doubles, the acceleration halves.

Key Idea

Newton’s Second Law in words

The acceleration of an object is proportional to the resultant force acting on it, and inversely proportional to the mass of the object.

Newton’s Second Law as an equation

The equation you need to recall and apply is:

F = ma

where:

$F$ is the resultant force in newtons, N
$m$ is the mass in kilograms, kg
$a$ is the acceleration in metres per second squared, m/s^2

Tip

Rearranging F = ma

If acceleration is unknown, use $\frac{F}{m}$ . If mass is unknown, use $\frac{F}{a}$ .

Example

Using F = ma with resistive forces

A car has a driving force of 2500 N forwards and resistive forces of 700 N backwards, so the resultant force is $\text{ N} - 700 \text{ N} = 1800 \text{ N}$ forwards.
The car’s mass is 1000 kg and acceleration is unknown, so use $\frac{F}{m}$ .
Substitute: $\frac{1800 \text{ N}}{1000 \text{ kg}} = 1.8 \text{ m/s}^2$ . The car accelerates forwards at 1.8 m/s^2.

Common Mistake

Using the wrong force

Do not put the engine thrust, pull or push into $F = ma$ unless it is already the resultant force. If friction, drag or another opposing force is given, combine the forces first.

Higher Tier: inertial mass

This part is Higher Tier only.

Definition

Inertial mass

Inertial mass is a measure of how difficult it is to change the velocity of an object. It is defined as the ratio of force over acceleration: $\frac{F}{a}$ .

A larger inertial mass means the same resultant force produces a smaller acceleration. This is why a loaded van is harder to get moving than an empty van.

Estimating forces in road transport

In exams, you may be asked to estimate speeds, accelerations and forces for everyday transport.

The symbol $∼\sim$ means “approximately” or “of the order of”. For example, a typical car might have $\sim 1000 \text{ kg}$ .

Useful rough values:

a cyclist and bike: about 100 kg
a small car: about 1000 kg to 1500 kg
motorway speed: about 30 m/s
a large car acceleration or braking acceleration: a few m/s^2

Example

Estimating a braking force

A 1200 kg car slows from 20 m/s to 0 m/s in 4.0 s, so $\frac{0 \text{ m/s} - 20 \text{ m/s}}{4.0 \text{ s}} = -5.0 \text{ m/s}^2$ .
Use Newton’s Second Law: $\text{ kg} \times -5.0 \text{ m/s}^2 = -6000 \text{ N}$ .
The negative sign means the force acts opposite to the motion. As an estimate, the braking force is $\sim 6000 \text{ N}$ backwards.

Required practical activity 7: investigating Newton’s Second Law

You need to know how to investigate:

the effect of varying the force on the acceleration of an object of constant mass
the effect of varying the mass on the acceleration produced by a constant force

Required practical setup with trolley, pulley, hanging masses and light gates

Varying force while keeping mass constant

To test how force affects acceleration:

Set up the trolley, string, pulley and hanging mass.
Keep the total moving mass constant.
Increase the driving force by moving slotted masses from the trolley to the hanging holder.
Release the trolley without pushing it.
Measure the acceleration using light gates or a data logger.
Repeat each reading and calculate a mean.

You should find that acceleration increases in direct proportion to resultant force.

Varying mass while keeping force constant

To test how mass affects acceleration:

Keep the hanging mass the same, so the driving force stays constant.
Add masses to the trolley to increase the total moving mass.
Measure the acceleration each time.
Repeat readings and calculate a mean.

You should find that increasing mass decreases acceleration.

The expected graph shapes are shown below.

Graphs showing acceleration against force, mass, and reciprocal mass

Common Mistake

Changing two variables at once

When testing the effect of force, keep the total moving mass constant by moving masses between the trolley and the hanging holder. Do not simply add extra masses to the hanger.

Exam technique

In the exam

Find the resultant force before using $F = ma$ .
Check units: force in N, mass in kg, acceleration in m/s^2.
For estimates, use rounded values and the symbol $∼\sim$ when your answer is approximate.
In practical questions, state what is changed, what is kept constant, and how acceleration is measured.

Self review

Check yourself

If a 2 kg object has a resultant force of 10 N, what is its acceleration?
Why does doubling the mass halve the acceleration when the resultant force stays the same?
In the trolley practical, how can you vary the force while keeping the total mass constant?

Newton's Second Law

What you'll learn

Start point: forces and resultant force

Acceleration: changing velocity

Newton’s Second Law in words

Newton’s Second Law as an equation

Higher Tier: inertial mass

Estimating forces in road transport

Required practical activity 7: investigating Newton’s Second Law

Varying force while keeping mass constant

Varying mass while keeping force constant

You've reached the end

Newton's Second Law

What you'll learn

Start point: forces and resultant force

Acceleration: changing velocity

Newton’s Second Law in words

Newton’s Second Law as an equation

Higher Tier: inertial mass

Estimating forces in road transport

Required practical activity 7: investigating Newton’s Second Law

Varying force while keeping mass constant

Varying mass while keeping force constant

You've reached the end

Newton's Second Law

What you'll learn

Start point: forces and resultant force

Acceleration: changing velocity

Newton’s Second Law in words

Newton’s Second Law as an equation

Higher Tier: inertial mass

Estimating forces in road transport

Required practical activity 7: investigating Newton’s Second Law

Varying force while keeping mass constant

Varying mass while keeping force constant

You've reached the end

Newton's Second Law Revision Guide

Newton's Second Law

What you'll learn

Start point: forces and resultant force

Acceleration: changing velocity

Newton’s Second Law in words

Newton’s Second Law as an equation

Higher Tier: inertial mass

Estimating forces in road transport

Required practical activity 7: investigating Newton’s Second Law

Varying force while keeping mass constant

Varying mass while keeping force constant

You've reached the end

Newton's Second Law Revision Guide