Part 1 · Forces, Scalars and Vectors

A force is a push or pull. Forces have units of Newtons (N). We measure force using a Newton meter.

Scalar quantities have size ("magnitude") only and no direction. Example: mass.

Scalar quantities can be added normally. Example: 36 kg + 14 kg = 50 kg.

Examples of scalars: mass, distance, speed, energy, time, power.

Vector quantities have both size (magnitude) and direction. Examples: force, velocity, acceleration, displacement, weight, momentum.

Part 2 · Scale Diagrams

In a scale diagram each force is drawn as an arrow of the appropriate length and direction (e.g. 1 cm = 1 N). Place the arrows head-to-tail. The resultant is the arrow from the start of the first to the end of the last.

Measure the resultant arrow length then multiply by the scale to find the magnitude of the resultant force.

Steps to find resultant from a scale drawing

• Step 1: Find the scale (1 cm = ? N)
• Step 2: Draw vectors head-to-tail at the correct scale
• Step 3: Measure the resultant arrow length (cm)
• Step 4: Multiply by the scale to get the resultant force (N)

Part 3 · Pythagoras' Theorem and Resultant Vectors

If one vector is at right angles to another, we can use Pythagoras' theorem to find the resultant vector.

We can also use scale diagrams to find a resultant vector.

In a scale diagram, draw each vector as an arrow to scale. Join them head-to-tail. The resultant is the arrow from start to finish.

The resultant vector is found by placing vectors head-to-tail and drawing an arrow from the tail of the first to the head of the last.

The direction of the resultant can be measured with a protractor from the scale diagram.

Part 1 · Gravity and Gravitational Field Strength

All objects attract each other due to the force of gravity. The strength of gravity at the surface of a planet is determined by its mass.

Gravitational field strength (g) is a measure of the gravitational force per kilogram at a location. On Earth, g = 9.8 N/kg.

Weight is the force on an object due to gravity. Weight is a vector, measured in Newtons (N).

Mass is the amount of matter an object contains. Mass is a scalar with SI unit kilogram (kg). Mass stays the same wherever the object is; weight changes with g.

The Moon has less mass than Earth, so its gravitational field strength is less (≈1.625 N/kg).

Part 2 · Calculating Weight: W = m × g

W = weight (N), m = mass (kg), g = gravitational field strength (N/kg). On Earth: g = 9.8 N/kg unless told otherwise.

Part 3 · Harder Calculations and Unit Conversions

When quantities are given with prefixes, convert to SI units first.

• Unit conversion: 1 kN = 1 000 N; 1 tonne = 1 000 kg; 1 g = 0.001 kg.
• An iPhone weighs 1.2 N on Earth: mass = 1.2 / 9.8 = 0.122 kg = 122 g.
• A car weighs 12 kN = 12 000 N: mass = 12 000 / 9.8 = 1 224 kg.

Part 1 · Force Diagrams

The forces acting on any object can be shown using a force diagram (free body diagram) — labelled arrows showing all forces acting on the object.

• The direction of each arrow shows the direction of the force
• The length of each arrow is proportional to the size (magnitude) of the force
• The resultant force is the single force that has the same effect as all forces combined
• To find the resultant: draw arrows tip to tail, then draw a line from start to finish

Part 2 · What the Resultant Means

• If there is a resultant force → the object is accelerating
• If there is no resultant force → no acceleration
• No acceleration means the object is either stationary OR moving at a constant velocity

Part 3 · Resultant Force with Perpendicular Vectors

When forces are perpendicular we can also look at them separately.

Sometimes there is no horizontal force. For example, if you throw an object (and air resistance is negligible) there is only gravity acting on it. Therefore it doesn't accelerate sideways at all — its horizontal velocity is constant.

If you shoot a bullet horizontally and drop a bullet at the same time, they will hit the ground at the same time — the sideways motion of the bullet doesn't matter as it is perpendicular to gravity.

Part 1 · Work Done = Energy Transferred

When a force moves an object, work is done. Work done equals the energy transferred. The units of work done are Joules (J).

W = work done (J), F = force applied (N), s = distance moved in the direction of the force (m). The force and distance must be in the same direction.

Part 2 · Finding Force and Distance

Part 3 · Questions with Unit Conversions

When quantities have prefixes, convert to SI units first before substituting into W = F × s.

Part 1 · Newton's First Law

For there to be acceleration, there must be an unbalanced (resultant) force. In other words, if the resultant force acting on an object is zero, its motion does not change.

There are two cases:

• If the object is stationary, it remains stationary
• If the object is moving, it continues to move at the same speed and in the same direction — so it continues at the same velocity

Part 2 · Newton's Second Law

When there is a resultant force, the object accelerates. The acceleration is directly proportional to the resultant force and inversely proportional to the mass.

F = resultant force (N), m = mass (kg), a = acceleration (m/s²). A bigger resultant force gives a bigger acceleration; a bigger mass gives a smaller acceleration.

Part 3 · Newton's Second Law Rearranged

Sometimes the resultant force is known and we need to find the mass or the acceleration instead. Always start from F = m × a, then rearrange to make the unknown the subject.

Remember to convert grams to kilograms first (1 g = 0.001 kg, so divide grams by 1 000).

Part 4 · Newton's Third Law

Whenever one object exerts a force on a second object, the second object exerts an equal and opposite force back on the first.

• These two forces are called an action–reaction pair — equal in size, opposite in direction, and the same type of force
• Importantly, the two forces act on different objects, so they never cancel each other out
• Example: a book pushes down on a table; the table pushes up on the book with an equal and opposite force

Part 5 · Newton's Laws Acting Together

Real situations involve several forces at once. To solve them, work in two steps.

• Step 1: find the resultant force. If it is zero the object is at rest or at constant velocity (Newton's First Law)
• Step 2: if the resultant force is not zero, use F = m × a with the resultant force (Newton's Second Law) to find the acceleration
• Remember to use the resultant force — not just one of the forces — and convert grams to kilograms first

Part 1 · The Forces on a Falling Object

When an object falls through the air, two forces act on it: its weight (downwards, caused by gravity) and air resistance (upwards, opposing its motion).

• The weight stays the same throughout the fall; air resistance increases as the object moves faster
• Just after the object starts to fall, air resistance is small, so weight is greater than air resistance — there is a resultant force downwards, so the object accelerates
• As it speeds up, air resistance increases, so the resultant force gets smaller and the acceleration decreases
• Eventually air resistance equals the weight. The resultant force is now zero, so there is no acceleration — the object falls at a steady speed called the terminal velocity

Part 2 · The Stages of a Skydive

• Stage A: The skydiver jumps from the plane. Weight pulls her down. Air resistance is very small at first, so she starts to accelerate
• Stage B: As she falls faster, air resistance increases and acts in the opposite direction to her weight
• Stage C: Air resistance continues to increase until it equals her weight. The forces are balanced, so she falls at a constant speed — her terminal velocity
• Stage D: She opens her parachute. There is now a very large upward air resistance force, so she decelerates (slows down)
• Stage E: As she slows down, air resistance decreases until it equals her weight again. She reaches a new, slower terminal velocity and lands safely

Part 3 · Velocity-Time Graphs

A velocity-time graph shows how the speed of a falling skydiver changes over time. Its shape tells the whole story of the jump.

• A steep, rising line means the skydiver is accelerating
• A horizontal line (zero gradient) means zero acceleration — a constant speed, or terminal velocity
• When the parachute opens, the line falls steeply (decelerating), then levels off at a lower, slower terminal velocity before reaching zero on landing
• Opening the parachute greatly increases the cross-sectional area, so air resistance increases sharply — air resistance equals the weight at a lower speed

Part 1 · Momentum: p = m × v

Any mass that is moving carries momentum.

p = momentum (in kg m/s), m = mass (in kg), v = velocity (in m/s). Because the equation contains velocity (a vector), momentum is also a vector: it acts in the direction the object is moving.

Part 2 · Rearranging: Finding Mass and Velocity

If the momentum is known, you can still find a missing mass or velocity — always starting from p = m × v.

• Write p = m × v, substitute in the values you know, and then rearrange to make the unknown the subject
• Do not learn a separate rearranged formula: substitute first, rearrange afterwards

Part 3 · Momentum with Unit Conversions

Always convert quantities to SI units before using p = m × v.

• Useful conversions: 1 tonne = 1 000 kg; 1 kg = 1 000 g; 1 km = 1 000 m; 1 mile ≈ 1 600 m
• To convert a speed in km/h to m/s, multiply by 1 000 then divide by 3 600
• If a speed is not given directly, find it from distance and time: v = s / t
• Remember to convert minutes and hours into seconds (1 minute = 60 s, 1 hour = 3 600 s)

Part 1 · The Law of Conservation of Momentum

Conservation of momentum means that the total momentum before a collision or explosion equals the total momentum afterwards (when no external forces act).

• This happens because the forces acting on each object during a collision are equal and opposite (Newton's Third Law)
• The momentum lost by one object is gained by the other, so total momentum is conserved
• Equation: total momentum before = total momentum after

Part 2 · Collision Calculations

For a collision where one object is initially at rest, work in two steps.

• Step 1: find the change in momentum of the moving object = (m × u) − (m × v)
• Step 2: this momentum is transferred to the second object. Its velocity = momentum gained ÷ its mass
• Always keep the units consistent (kg, m/s, kg m/s)

Part 3 · Explosions and Recoil

Conservation of momentum also applies to explosions (objects flying apart).

• Before an explosion, total momentum = 0 (both objects at rest)
• After the explosion, the two objects move in opposite directions with equal and opposite momenta, so the total is still zero
• Example: firing a rifle. The bullet moves forward; the rifle recoils backward
• A bullet (0.01 kg) fired from a rifle (2 kg) that recoils at 1 m/s: momentum of rifle = 2 × 1 = 2 kg m/s, so momentum of bullet = 2 kg m/s; v_bullet = p / m = 2 / 0.01 = 200 m/s

Part 1 · Thinking Distance

Thinking distance is the distance the vehicle travels while the driver reacts to a hazard — from seeing the hazard to pressing the brake — before the brakes are applied.

• Reaction time is the time taken to react. For a healthy, alert driver it is typically 0.2–0.9 s
• Thinking distance is calculated from: thinking distance = speed × reaction time
• Thinking distance is therefore directly proportional to speed: double the speed and the thinking distance doubles
• Factors that increase the thinking distance by increasing the reaction time: tiredness, alcohol, drugs (including some medicines) and distractions (such as using a phone or loud music)

Part 2 · Braking Distance

Braking distance is the distance the vehicle travels from the moment the brakes are applied to the moment it stops.

• During braking, the kinetic energy of the vehicle is transferred to thermal (heat) energy in the brakes by the work done against friction — the brakes, tyres and road warm up
• Braking distance is proportional to the speed squared (v²): doubling the speed makes the braking distance about four times larger, so a braking-distance–speed graph is a curve
• Factors that increase the braking distance: a higher speed; adverse weather conditions (rain, ice or snow); poor road conditions (a wet, icy or loose/greasy surface); and a vehicle in poor condition (worn brakes or worn tyres)
• Adverse weather and road conditions reduce the grip (friction) between the tyres and the road, so the vehicle travels further before stopping. A more heavily loaded (greater mass) vehicle also has a longer braking distance

Part 3 · Stopping Distance

• It is the total distance travelled from the moment the driver sees a hazard to the moment the vehicle stops
• When the speed doubles, the thinking distance doubles (proportional to speed) but the braking distance increases about four-fold (proportional to v²), so the stopping distance grows quickly with speed
• Adverse conditions increase the stopping distance: e.g. tiredness or alcohol increases the thinking distance, while rain or ice increases the braking distance