Newton’s Laws and Forces Explained

Introduction - Kinematics and Motion Simplified

Think about when you go for a walk. Sometimes you will walk fast sometimes. Sometimes you need to slow down in order to stop. In both cases, you cover a specific distance and your speed will change. This is motion and the area of maths behind it is kinematics.

Having a thorough understanding of kinematics is essential for A Level Maths as it will be used to look at objects moving in a straight line, along slopes and also projectiles.

In this guide we will talk about SUVAT, graphs of motion, acceleration and also projectiles.

Essential Concepts

There are some terms that you need to be aware which are as follows:

Displacement (s): The straight-line distance from start to finish, including direction.
Distance: Total distance travelled.
Speed (v): Distance travelled divided by time.
Velocity (v): Total displacement divided by time (vector).
Acceleration (a): The rate of change of velocity.

Example: A car travels 100 m east in 10 s.

Distance = 100 m
Displacement = 100 m east
Average speed = 100 ÷ 10 = 10 m/s

Average velocity = 10 m/s east

F = ma is Newton's Second Law.

Newton’s Second Law shows a relationship between mass, acceleration, and force:

$F = ma$

F = the net force in Newtons (N)
m = mass (in kilogrammes, kg)
a = acceleration (m/s²)

Example: An object that weighs 3 kg speeds up at 4 m/s². The force = $3 \times 4 = 12$ N.

Applications:

How fast do cars go?.
Rockets blasting off to space. The greater the mass the greater the force.
Falling objects due to gravity.

For any exam questions it is important to do the following:

Draw a free-body diagram.
If forces are at an angle, write them in components.
Apply F = ma.

SUVAT Equations

When motion is moving with constant acceleration in a straight line, we use SUVAT equations to solve problems. Even though you get the SUVAT equations in a formula booklet you should really try to remember them.

S – displacement (m)
U – initial velocity (m/s)
V – final velocity (m/s)
A – acceleration (m/s²)
T – time (s)

The main equations:

$v = u + a t$
$s = u t + \tfrac12 a t^2$
$s = v t – \tfrac12 a t^2$
$v^2 = u^2 + 2 a s$
$s = \tfrac{(u + v)}{2}\,t$

Worked Example:
Find the acceleration of a car that accelerates from 5 m/s to 25 m/s in 10 s.

Use $v = u + a\,t$

25 = 5 + a \times 10

25 – 5 = 10\,a

20 = 10\,a

a = \frac{20}{10}

a = 2\ \mathrm{m/s}^2

Displacement–Time Graphs

With a displacement–time graph, this shows how the position of an object changes over time.

The slope (or gradient) gives velocity.
A flat line means stationary.
A steep line (or gradient) indicates faster motion.
A curved line shows changing velocity (acceleration).

Example:
A train moves 50 m in 5 s, then stops for 2 s, then moves 30 m in 3 s. The graph shows straight lines for constant motion and flat sections for stops.

Tips:

Always label both axes: time (x-axis), displacement (y-axis).
Check the units for slope: m/s.

Here’s the distance–time graph for the train’s motion — showing two constant-motion segments (sloped lines) and a flat section while the train stops.

Velocity–Time Graphs

A Velocity–time graph shows how velocity changes over time.

The slope = acceleration.
The area under the curve = total distance travelled.

Example: A car accelerates from 0 to 20 m/s in 10 s:

You will find that the graph is a straight line. So finding the acceleration is just the gradient of the line.

Acceleration = ( 20 ÷ 10 = 2 m/s² )
The distance travelled is the area under the curve:

The distance travelled $s$ is the area under the curve.

The base of the triangle is $10$ seconds.

The height of the triangle is $20$ m/s.

So the distance travelled can be calculated as:

s \;=\;\tfrac12\times(\text{base})\times(\text{height}) = \tfrac12 \times 10 \times 20 = 100\ \text{m}.

Practical tip: If the shape under the curve is made up of compound shapes, break the area into shapes that you can easily work.

Acceleration and Motion

When it comes to acceleration, it can be constant or variable.

Constant acceleration: Use the SUVAT equations.
Variable acceleration: Use calculus as needed.

Free fall example: Objects falling under gravity:

Acceleration ( a = 9.8 m/s² ) downwards
Ignore air resistance for basic problems

Example: Drop a ball from 20 m:

Using the SUVAT equation:

$v^2 = u^2 + 2as$

Since the ball is dropped:

Initial speed $u = 0$
Acceleration $a = 9.8,\text{m/s}^2$
Distance fallen $s = 20,\text{m}$

So:

$v^2 = 0^2 + 2(9.8)(20)$

$v^2 = 392$

Remember to square root to find the speed:

$v = \sqrt{392} \approx 19.8,\text{m/s}$

Motion of a Projectile

With projectiles you have motion that is horizontal and vertical:

Horizontal velocity $u_x = u \cos\theta$ is constant as it does not depend on gravity.
Vertical velocity $u_y = u \sin\theta$ is dependent on gravity.
Always use SUVAT equations horizontally and vertically in each direction.

Example: A ball is thrown at 10 m/s at 30°:

Horizontal: ux = 10 cos 30° = 5√3 ≈ 8.66 m/s
Vertical: uy = 10 sin 30° = 5 m/s

Time to peak: 5/9.8 ≈ 0.51 s

Max height: 25/(19.6) ≈ 1.28 m

Range: (100·(√3/2))/9.8 ≈ 8.84 m

Practical Problem-Solving Tips

Always draw a clear, large diagram.
Identify all known quantities: s, u, v, a, t.
Choose the correct SUVAT equation.
Check your units consistently.
For motion at an angle, resolve into horizontal and vertical components.
Sometimes using a displacement–time and velocity–time graphs can be easier and quicker than using SUVAT.

If you are looking for more guidance on SUVAT and mechanics in general then join our A Level Maths revision course. We look at more challenging examples and further question set on SUVAT and motion problems.

Conclusion

Kinematics is all about understanding motion.

For motion under constant acceleration use SUVAT.
Read any graphs carefully to interpret displacement, velocity, and acceleration.
When working with motion at angle, split into horizontal and vertical components.

Keep practising these concepts to build confidence and speed for your A-Level exams. Having a deep understanding of kinematics will prepare you for more advanced topics like projectiles.

About the Author – S. Mahandru

About the Author – S. Mahandru

S. Mahandru is Head of Maths at Exam.tips. With 15+ years of teaching experience, he helps students make sense of A-Level and GCSE maths. He creates clear guides, worked examples, and revision courses to boost confidence and exam success.

🧭 Next topic:

“Next, see how Newton’s Laws explain the forces behind the motion you’ve just learned.”

FAQS

What are the five SUVAT equations in kinematics?

The five SUVAT equations describe motion under constant acceleration. They relate displacement (s), initial velocity (u), final velocity (v), acceleration (a), and time (t).
They are:

$v = u + a\,t$
$s = u\,t + \tfrac{1}{2}a\,t^2$
$v^2 = u^2 + 2\,a\,s$
$s = \tfrac{1}{2}(u + v)\,t$
$s = v\,t – \tfrac{1}{2}a\,t^2$

These equations are essential for solving A Level Maths and Mechanics problems.

What’s the difference between speed and velocity?

Speed is a scalar quantity that measures how fast an object moves, regardless of direction.
Velocity is a vector quantity, meaning it includes both magnitude and direction.
For example, a car travelling at 10 m/s east has a velocity of 10 m/s east — but just a speed of 10 m/s.

How do I know which SUVAT equation to use in a question?

Write down all the quantities you’re given and what you need to find (s, u, v, a, t).
Then, choose the equation that contains only those variables.
For example, if you know u, v, and a, and need s, use $v^2 = u^2 + 2\,a\,s$ .
Always check your units and use a clear diagram — this helps prevent mistakes in exams.