Equations of Motion

2026 Syllabus Objectives

By the end of this topic, you should be able to:

  1. Define and use distance, displacement, speed, velocity and acceleration
  2. Use graphical methods to represent distance, displacement, speed, velocity and acceleration
  3. Determine displacement from the area under a velocity–time graph
  4. Determine velocity using the gradient of a displacement–time graph
  5. Determine acceleration using the gradient of a velocity–time graph
  6. Derive, from the definitions of velocity and acceleration, equations that represent uniformly accelerated motion in a straight line
  7. Solve problems using equations that represent uniformly accelerated motion in a straight line, including the motion of bodies falling in a uniform gravitational field without air resistance
  8. Describe an experiment to determine the acceleration of free fall using a falling object
  9. Describe and explain motion due to a uniform velocity in one direction and a uniform acceleration in a perpendicular direction

1. Distance, Displacement, Speed, Velocity and Acceleration

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Distance and Displacement

Distance is a measure of how far an object has travelled. It only tells you the total length of the path taken, regardless of direction. Distance is a scalar quantity, which means it has size (magnitude) but no direction.

Displacement is the distance moved in a particular direction. It tells you how far the object is from its starting position, measured in a straight line. Displacement is a vector quantity, which means it has both size and direction.

Example: Imagine you walk 7 km north, then 8 km east. Your total distance travelled is 15 km (7 + 8). However, your displacement might only be 10 km in a north-easterly direction (the straight-line distance from start to finish).

Units: Both distance and displacement are measured in metres (m) in the SI system.

Speed and Velocity

Speed tells you how fast something is moving. It is the distance travelled per unit time. Speed is a scalar quantity (no direction).

The formula for average speed is:

average speed = distance travelled ÷ time taken

or in symbols: v = d/t

Instantaneous speed is the speed at a specific moment in time (like looking at a car's speedometer).

Velocity is speed in a particular direction. It is the rate of change of displacement. Velocity is a vector quantity.

The formula for velocity is:

velocity = displacement ÷ time taken

or in symbols: v = s/t or v = Δs/Δt

(The symbol Δ means "change in")

Units: Speed and velocity are measured in metres per second (m s⁻¹) in the SI system. Other common units include km h⁻¹ (kilometres per hour) and mph (miles per hour).

Example: A car travels at 15 m s⁻¹ for 1 hour (3600 seconds). How far does it travel?

Using s = v × t: s = 15 × 3600 = 54,000 m = 54 km

Acceleration

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

The formula for average acceleration is:

acceleration = change in velocity ÷ time taken

or in symbols: a = (v − u)/t or a = Δv/Δt

where:

  • u = initial velocity
  • v = final velocity
  • t = time taken

Acceleration is a vector quantity and is measured in metres per second squared (m s⁻²).

Deceleration means negative acceleration (slowing down).

Important note: If an object changes direction, it is accelerating, even if its speed stays constant. This is because velocity is a vector – if the direction changes, the velocity changes.

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