Speed, Velocity and Acceleration

Describing motion

To describe how things move we use distance, displacement, speed, velocity and acceleration. Some are scalars (size only) and some are vectors (size and direction).

• Scalars: distance, speed. Vectors: displacement, velocity, acceleration, force.

Speed and velocity

• Speed is how fast an object moves (a scalar). Velocity is speed in a given direction (a vector).

speed = distance ÷ time      (v = s ÷ t)

Rearrange: distance = speed × time, time = distance ÷ speed.

Typical speeds to know: walking ≈ 1.5 m/s, running ≈ 3 m/s, cycling ≈ 6 m/s. The speed of sound in air ≈ 330 m/s.

Acceleration

Acceleration is the rate of change of velocity — how quickly velocity changes (a vector). Measured in m/s².

acceleration = change in velocity ÷ time   (a = Δv ÷ t)

A negative acceleration means decelerating (slowing down).

Uniform acceleration equation:

v² = u² + 2as

(v = final velocity, u = initial velocity, a = acceleration, s = distance).

Objects in free fall near Earth accelerate at g ≈ 9.8 m/s² due to gravity.

Distance–time graphs

• Gradient (slope) = speed.
• A horizontal line = stationary (not moving).
• A steeper line = faster; a curved line = changing speed (accelerating).

Velocity–time graphs

• Gradient = acceleration (a flat line = constant velocity; a steeper slope = greater acceleration).
• Area under the graph = distance travelled.

Worked example

A car accelerates from 10 m/s to 30 m/s in 5 seconds. Find the acceleration.

a = (v − u) ÷ t = (30 − 10) ÷ 5 = 20 ÷ 5 = 4 m/s²

✓

Common mistakes

• Confusing speed (scalar) with velocity (vector, has direction).
• Mixing up the graphs — on a distance–time graph the gradient is speed; on a velocity–time graph the gradient is acceleration and the area is distance.
• Forgetting units: acceleration is m/s², not m/s.

Exam tips

• Learn which quantities are scalars vs vectors.
• Know how to read both graph types (gradient and area).
• Practise rearranging v = s÷t and a = Δv÷t, and using v² = u² + 2as.

Key facts to remember

• Speed = distance ÷ time; velocity = speed with direction; acceleration = change in velocity ÷ time (m/s²).
• Distance–time graph: gradient = speed. Velocity–time graph: gradient = acceleration, area = distance.
• Free-fall acceleration g ≈ 9.8 m/s²; v² = u² + 2as for uniform acceleration.