Speed, Velocity and Acceleration
The definitions, formulas and typical calculations for motion in a straight line.
Speed and velocity
Speed is how fast something moves, regardless of direction. Velocity is speed in a given direction, so velocity is a vector and speed is a scalar.
speed = distance ÷ time
Units: metres per second (m/s) or kilometres per hour (km/h).
Example. A cyclist travels 120 m in 20 s. Speed = 120 ÷ 20 = 6 m/s.
To convert m/s to km/h, multiply by 3.6. So 6 m/s = 21.6 km/h.
Acceleration
Acceleration is the rate of change of velocity:
a = (v − u) ÷ t
where u is the initial velocity, v the final velocity and t the time.
Example. A car speeds up from 5 m/s to 25 m/s in 8 s. a = (25 − 5) ÷ 8 = 2.5 m/s².
A negative acceleration means slowing down (deceleration), if the object is moving in the positive direction.
Equations of motion at constant acceleration
- v = u + at
- s = ut + ½at²
- v² = u² + 2as
Example. A ball is dropped from rest. Taking g = 9.8 m/s², find its speed after 3 s.
- v = u + at = 0 + 9.8 × 3 = 29.4 m/s.
Distance–time and velocity–time graphs
- On a distance–time graph, the gradient is speed. A horizontal line means stationary.
- On a velocity–time graph, the gradient is acceleration and the area under the graph is distance travelled.
Common mistakes
- Using minutes with metres per second without converting.
- Confusing distance (total path length) with displacement (straight-line distance in a direction).
- Saying a negative acceleration always means "slowing down". It depends on the direction of motion.
Quick recap
Speed = distance ÷ time; acceleration = change in velocity ÷ time. Use gradients and areas on graphs.