Forces, Newton's Laws & Momentum

Forces as pushes and pulls

A force is a push or a pull that one object exerts on another. Forces are measured in newtons (N) and are vector quantities, meaning they have both a size (magnitude) and a direction. To describe a force fully you must state both: "20 N downwards" is a force; "20 N" on its own is incomplete.

Because forces are vectors, we draw them as arrows. The length of the arrow shows the size of the force and the way it points shows its direction.

Key terms

Force — a push or pull, measured in newtons (N).

Vector — a quantity with both magnitude and direction (e.g. force, velocity, momentum).

Scalar — a quantity with magnitude only (e.g. mass, speed, energy).

Types of force

You should be able to recognise these common forces:

Weight — the pull of gravity on a mass, always acting downwards.
Normal (reaction) force — the push of a surface on an object, at right angles to the surface.
Friction — a force that opposes motion between two surfaces in contact.
Air resistance (drag) — friction from a fluid (air or water), opposing motion through it.
Tension — the pull in a stretched rope, string or spring.
Thrust — the driving force from an engine or motor.
Upthrust — the upward push of a fluid on an object floating or submerged in it.

Free-body force diagrams

A free-body force diagram shows a single object as a dot or box, with arrows for every force acting on it. The arrows start at the object and point in the direction of each force. Longer arrows mean bigger forces.

Resultant force

When several forces act on an object, their combined effect is the resultant force — a single force that has the same effect as all of them together.

For forces along a straight line, simply add forces in the same direction and subtract those that oppose. If a 600 N thrust acts forward and 200 N of drag acts backward, the resultant is $600 - 200 = 400\,\text{N}$ forward.

If all the forces cancel out, the resultant is zero and we say the forces are balanced.

Newton's first law

Newton's first law: an object stays at rest, or keeps moving at constant velocity, unless acted on by a resultant (unbalanced) force.

This means a zero resultant force does not mean "no motion" — it means "no change in motion". A car cruising at a steady 30 m/s on a straight road has balanced forces (thrust = drag, normal = weight), even though it is moving fast.

A change in velocity is an acceleration. So if forces are balanced, acceleration is zero. To speed up, slow down, or change direction, you need an unbalanced resultant force.

Exam tip

"Constant velocity" and "at rest" both mean zero resultant force. If a question says an object moves at steady speed in a straight line, you can immediately write that the forces are balanced.

Newton's second law: F = ma

When there is a resultant force, the object accelerates in the direction of that force. The bigger the force, the bigger the acceleration; the bigger the mass, the smaller the acceleration. This is written:

$$F = m \times a$$

where $F$ is the resultant force in newtons (N), $m$ is mass in kilograms (kg) and $a$ is acceleration in metres per second squared (m/s²).

Worked example

A trolley of mass $4\,\text{kg}$ is pushed with a resultant force of $12\,\text{N}$. Find its acceleration.

Rearrange $F = ma$ to give $a = \dfrac{F}{m}$.

Newton's third law

Newton's third law: for every action there is an equal and opposite reaction.

If object A pushes on object B, then object B pushes back on A with a force that is equal in size but opposite in direction. The two forces always act on different objects.

For example, when you jump, your feet push down on the Earth and the Earth pushes back up on you with an equal force — that reaction force launches you upward. A rocket pushes gas downwards and the gas pushes the rocket upwards.

Watch out

A third-law pair always acts on two different objects. The weight of a book and the normal force from the table are not a third-law pair — they both act on the book, and are balanced forces (first law), not action–reaction.

Weight, mass and W = mg

Mass is the amount of matter in an object, measured in kilograms (kg). It is the same everywhere in the universe.

Weight is the force of gravity pulling on that mass, measured in newtons (N). Weight depends on the strength of the gravitational field:

$$W = m \times g$$

where $g$ is the gravitational field strength. On Earth $g \approx 10\,\text{N/kg}$ (some questions use $9.8$).

Worked example

An astronaut has a mass of $70\,\text{kg}$. Find her weight on Earth ($g = 10\,\text{N/kg}$).

$W = mg = 70 \times 10 = 700\,\text{N}$.

Terminal velocity

When an object falls through air, two forces act on it: its weight (down) and air resistance / drag (up). Drag increases as the object speeds up.

1. At the start, drag is small, so weight is much bigger. The resultant force is large and downwards, so the object accelerates.
2. As it speeds up, drag grows, so the resultant force shrinks and the acceleration falls.
3. Eventually drag grows until it equals weight. The resultant force is now zero, so acceleration is zero. The object falls at a steady, maximum speed — the terminal velocity.

Momentum

The momentum of a moving object measures how hard it is to stop. It depends on mass and velocity:

$$p = m \times v$$

where $p$ is momentum in kilogram metres per second (kg m/s), $m$ is mass in kg and $v$ is velocity in m/s. Momentum is a vector — direction matters.

Key terms

Momentum — mass × velocity, in kg m/s; a vector quantity.

Conservation of momentum — in a closed system with no external forces, total momentum before = total momentum after.

Conservation of momentum in collisions

In any collision (or explosion), the total momentum before equals the total momentum after, provided no external force acts. You add up momentum of all objects before, and set it equal to the total after — remembering that momentum in opposite directions has opposite signs.

Worked example

A $2\,\text{kg}$ trolley moving at $3\,\text{m/s}$ hits a stationary $2\,\text{kg}$ trolley and they stick together. Find their combined speed.

Momentum before $= (2 \times 3) + (2 \times 0) = 6\,\text{kg m/s}$.

Force as rate of change of momentum

A resultant force changes an object's momentum. Newton's second law can be written as:

$$F = \frac{\Delta p}{\Delta t} = \frac{m\,\Delta v}{\Delta t}$$

In words: force equals the rate of change of momentum. This shows that the same change in momentum can be produced by a large force over a short time, or a small force over a long time.

Safety features and momentum

When a car crashes, the passengers undergo a large change in momentum. From $F = \Delta p / \Delta t$, the longer the time taken to lose that momentum, the smaller the force on the passengers — and a smaller force means fewer injuries.

Crumple zones in the car body fold and deform on impact, extending the collision time.
Seat belts stretch slightly, letting the body slow down over a longer time.
Air bags cushion the head, increasing the stopping time and spreading the force.

Exam tip

For "explain the safety feature" questions, always link it back to the equation: the feature increases the time for the change in momentum, so $F = \Delta p / \Delta t$ gives a smaller force, reducing injury. Quoting the equation gains marks.