Motion, Forces & Energy

Measurement, scalars and vectors

Physics begins with measurement. Before any law can be written, a quantity has to be pinned to a number and a unit. Two questions decide how that number behaves: how big is it, and does it point somewhere?

Measuring the basics

Every measurement uses the right instrument for the size of the quantity. Length is read with a ruler or measuring tape; a long path can be measured with a trundle wheel. The volume of a liquid is read from a measuring cylinder at the bottom of the meniscus. Time is taken with a stopwatch, though a human reaction time of about 0.3 s limits short measurements.

Accuracy improves by repeating and averaging, and by measuring multiples of a small quantity. To find the volume of an irregular solid, lower it into water and read the rise — the displacement method.

Scalars and vectors

A scalar is fully described by a number and a unit: distance, speed, mass, time, energy and temperature are all scalars. A vector also needs a direction: force, weight, velocity, acceleration and momentum are vectors. Telling the two apart is a recurring one-mark question.

Two vectors that act at an angle combine into a single resultant. When they are perpendicular, the resultant is found by drawing them tip-to-tail and completing the right-angled triangle — its length gives the magnitude and the angle gives the direction.

Worked example — resultant force

A box is pulled by a force of 4.0 N east and 3.0 N north. The forces are perpendicular, so the magnitude comes from Pythagoras: R = √(4.0² + 3.0²) = √25 = 5.0 N. The direction from the east axis is tan θ = 3.0 / 4.0, so θ = 37° north of east. A single 5.0 N force at 37° has exactly the same effect as the two original forces together.

Speed, velocity, acceleration and motion graphs

Motion is change of position with time. The same journey can be described by how fast you go, by where that speed points, and by how quickly the speed itself is changing.

Speed and velocity

Speed measures how much ground is covered each second. Most journeys vary, so we usually quote the average speed — total distance divided by total time, v = s / t. Velocity is speed with a direction attached; a car rounding a bend at a steady 30 m/s has constant speed but changing velocity, because its direction keeps turning.

Worked example: a sprinter covers 100 m in 12.5 s, so the average speed is v = s / t = 100 / 12.5 = 8.0 m/s. This is the average over the whole race; the sprinter’s top speed near the finish is higher.

Acceleration

An object accelerates whenever its velocity changes — speeding up, slowing down, or changing direction. Every candidate should be able to recognise acceleration from a graph or a description.

Acceleration is the rate of change of velocity: a = Δv / t, in m/s². It is a vector. A car speeding up from rest to 20 m/s in 8.0 s has a = 20 / 8.0 = 2.5 m/s² — it gains 2.5 m/s of velocity every second it accelerates.

Free fall and terminal velocity

Close to Earth, a falling object accelerates at g = 9.8 m/s². As it speeds up, air resistance grows. At first the weight is much larger than the drag, so the object accelerates. The drag rises with speed until it equals the weight; the resultant force is then zero, acceleration stops, and the object falls at a steady terminal velocity. A skydiver shows the full sequence: accelerate after jumping, reach terminal velocity, then — once the parachute opens — decelerate to a new, lower terminal velocity for a safe landing.

Distance–time graphs

On a distance–time graph, the gradient is the speed. A horizontal line means the object is not moving; a straight slope means a steady speed; a steeper slope means a higher speed. A curve that gets steeper shows an object speeding up. "Describe the motion" wants the shape in words, not a number.

ExtendedSpeed–time graphs

On a speed–time graph, a horizontal line is constant speed and a sloping line is acceleration. Reading these shapes is Core.

Two quantities are hidden in the graph. The gradient of the line gives the acceleration. The area between the line and the time axis gives the distance travelled. For a curved line, the instantaneous acceleration is the gradient of the tangent at that point.

Worked example: a train accelerates from rest to 30 m/s in 20 s, holds 30 m/s for 40 s, then stops in 10 s. The distance is the area under the graph — a triangle, a rectangle and a triangle: (½ × 20 × 30) + (40 × 30) + (½ × 10 × 30) = 300 + 1200 + 150 = 1650 m.

Mass, weight and density

Mass and weight are confused more often than any other pair in physics. One counts the matter in an object; the other is the pull of gravity on it. Density ties them to size.

Mass and weight

Mass measures how much matter an object contains and does not change when the object is moved. Weight is the gravitational force acting on that mass, so it depends on the local gravitational field strength g. On Earth g = 9.8 N/kg; on the Moon it is only about 1.6 N/kg, so the same astronaut weighs far less there while keeping the same mass. The relationship is W = mg, and rearranged g = W / m — which is why gravitational field strength can be defined as the weight per unit mass.

Density and floating

Density compares an object’s mass with the space it fills. To measure it, find the mass on a balance and the volume — by measuring the dimensions of a regular solid, or by the displacement method for an irregular one — then divide. An object floats if its density is less than the fluid’s, and sinks if it is greater.

Worked example: a metal block has a mass of 200 g and a volume of 25 cm³. Its density is ρ = m / V = 200 / 25 = 8.0 g/cm³ = 8000 kg/m³. This is denser than water, so the block would sink.

Forces and Newton’s laws

A force is a push or a pull. It can change an object’s speed, its direction, or its shape. Newton’s three laws explain exactly how forces and motion are linked.

Resultant force and the first law

When several forces act, they add to a single resultant. If the forces are balanced the resultant is zero; if unbalanced, the resultant changes the motion. Newton’s first law: an object stays at rest, or moves at constant velocity, unless a resultant force acts on it. This tendency to keep doing what it is doing is called inertia.

ExtendedThe second and third laws

Newton’s second law makes the link exact: resultant force = mass × acceleration, F = ma, with the acceleration in the direction of the resultant force. The larger the resultant force, the larger the acceleration; the larger the mass, the smaller the acceleration. A resultant force of 600 N on a 1500 kg car gives a = F / m = 600 / 1500 = 0.40 m/s².

Newton’s third law says forces come in pairs: if A pushes B, then B pushes A with an equal force in the opposite direction, and the two forces act on different objects. A rocket pushes gas down; the gas pushes the rocket up.

ExtendedHooke’s law

When a spring is stretched, its extension (the increase in length) grows in step with the force applied — double the load, double the extension — until the limit of proportionality is passed, beyond which extension is no longer proportional to load.

The constant of proportionality is the spring constant k: F = kx. A force of 2.0 N that stretches a spring by 0.050 m gives k = F / x = 2.0 / 0.050 = 40 N/m.

Circular motion and friction

An object moving in a circle at steady speed is still accelerating, because its direction — and therefore its velocity — is always changing. A resultant force, the centripetal force, acts toward the centre. The force needed is larger for a higher speed or greater mass, and smaller for a larger radius.

Friction is a force that opposes the relative motion of two surfaces in contact. Air resistance (drag) is friction with the air, and grows as an object moves faster — the effect behind terminal velocity. Friction is often wasteful, transferring kinetic energy to the thermal store as heat, but it is also essential: without it brakes could not grip and walking would be impossible.

Moments and centre of gravity

A force applied away from a pivot turns it. The size of that turning effect — the moment — explains seesaws, spanners, cranes and why a door handle sits as far as possible from the hinge.

The moment of a force

The moment of a force measures its turning effect about a pivot. It grows with both the size of the force and its perpendicular distance from the pivot — which is why a long spanner loosens a tight bolt that fingers cannot. moment = F × d.

The principle of moments

A balanced beam is in equilibrium: there is no resultant force and no resultant turning effect, so the clockwise and anticlockwise moments about the pivot must be equal.

Worked example: a 300 N weight sits 2.0 m left of a pivot. The weight W needed 1.5 m to the right to balance it comes from equal moments: W × 1.5 = 300 × 2.0 = 600 N·m, so W = 600 / 1.5 = 400 N.

Centre of gravity

The centre of gravity is the single point at which the whole weight of an object can be taken to act. For an irregular card, it is found by hanging the card from a pin, drawing a vertical from a plumb line, and repeating from another point — the lines cross at the centre of gravity. An object is more stable when its centre of gravity is low and its base is wide.

Momentum and impulse

The whole of this section is Extended — examined on Papers 2 and 4 only. Momentum measures how hard a moving object is to stop. It combines mass and velocity into one quantity that, in a closed system, is never lost — only passed between objects.

ExtendedMomentum and impulse

A heavy lorry and a fast bullet are both hard to stop because each has large momentum. Momentum is the product of mass and velocity, p = mv, and because velocity is a vector, so is momentum. A resultant force changes momentum, and the impulse of a force, FΔt, equals the change in momentum it causes.

ExtendedConservation of momentum

When two objects interact and no outside force acts, the total momentum is conserved: whatever momentum one object gains, the other loses.

Worked example: a 2.0 kg trolley at 3.0 m/s strikes and sticks to a stationary 1.0 kg trolley. Momentum before is (2.0 × 3.0) + (1.0 × 0) = 6.0 kg·m/s. After, the combined 3.0 kg mass moves at v where 6.0 = 3.0 × v, so v = 2.0 m/s in the original direction.

Energy, work and power

Energy is the currency of physics. It is never made or destroyed — only moved from one store to another. Track where it goes and you can predict what happens next.

Stores, transfers and conservation

Energy is held in stores and moved along transfer pathways. A battery’s chemical store drives a current (electrical) that heats a wire (heating). Throughout, the principle of conservation of energy holds: the total amount of energy stays the same — it is only shifted between stores. A falling ball is the clearest case: its gravitational potential store empties into a kinetic store as it speeds up, the total staying constant until it hits the ground.

ExtendedKinetic and potential energy

Kinetic energy is the energy an object has because it is moving: Eₖ = ½mv², in joules. The change in gravitational potential energy when an object is raised depends on its weight and the height gained: ΔEₚ = mgΔh.

Worked example: a 1000 kg car travelling at 20 m/s has Eₖ = ½ × 1000 × 20² = ½ × 1000 × 400 = 200 000 J = 200 kJ.

Work and power

Work is energy transferred by a force, measured in joules — the same unit as energy. The distance used must be measured in the direction of the force. Power is how quickly that energy is transferred; a more powerful motor does the same work in less time. A motor that transfers 6000 J in 3.0 s has a power of P = E / t = 6000 / 3.0 = 2000 W = 2.0 kW.

ExtendedEfficiency and energy resources

Efficiency is the fraction of input energy transferred usefully: efficiency = useful output / total input (× 100 for a percentage). It is always less than 100% — some energy is always wasted, usually as heat. Of 100 J supplied to a device that delivers 60 J usefully, 40 J is wasted as heat, an efficiency of 60%.

Renewable resources — solar, wind, hydroelectric, tidal, wave, geothermal and biofuel — will not run out. Non-renewable resources — fossil fuels and nuclear fuel — are finite. Each is a trade-off between reliability, cost and environmental impact.

Pressure

The same force feels very different spread over a wide area or concentrated on a point. Pressure captures that difference — and explains why knives are sharp and snowshoes are wide.

Pressure on a surface

Pressure is how concentrated a force is. A sharp knife or a drawing pin puts a small force over a tiny area, giving a very high pressure; snowshoes spread weight over a large area, giving a low pressure that stops the wearer sinking. A box pushing down with 600 N over 0.020 m² gives p = F / A = 600 / 0.020 = 30 000 Pa = 30 kPa.

ExtendedPressure in liquids

Liquid pressure rises with depth and acts equally in all directions at a given depth. It does not depend on the container’s shape.

In a liquid, pressure increases with depth because a deeper point supports a taller column of liquid above it. The extra pressure depends on depth, density and gravity: Δp = ρgΔh.