01 · 20 min
Motion
Speed, velocity, acceleration, and the equations for constant acceleration.
v = u + at
The idea
Speed is distance travelled divided by time. Velocity is speed in a stated direction, so it is a vector. Average speed uses the total distance, even if the object turns around.
Acceleration is how quickly velocity changes. A car that speeds up, slows down, or turns has an acceleration. For motion in a straight line with steady acceleration, three equations connect u, v, a, s, and t.
On a velocity–time graph the slope is acceleration and the area under the line is displacement. A horizontal line means the velocity is not changing.
| Shape | Meaning |
|---|---|
| Horizontal line on the time axis | Stationary |
| Horizontal line above the axis | Constant velocity |
| Straight sloping line | Constant acceleration |
| Steeper slope | Larger acceleration |
| Area between the line and the time axis | Displacement |
Formulas
Average speed
v = s / t
Use for a whole journey. s is distance travelled. It is not a vector.
- v
- average speed
- m/s
- s
- distance
- m
- t
- time
- s
Acceleration
a = (v − u) / t
Change in velocity over time. A negative value means a deceleration if forward is positive.
- a
- acceleration
- m/s²
- v
- final velocity
- m/s
- u
- initial velocity
- m/s
- t
- time
- s
Velocity after time
v = u + at
Constant acceleration in a straight line. Same relation as the definition of a.
- v
- final velocity
- m/s
- u
- initial velocity
- m/s
- a
- acceleration
- m/s²
- t
- time
- s
Displacement
s = ut + (1/2)at²
How far it travels under constant acceleration. From rest, this becomes s = (1/2)at².
- s
- displacement
- m
- u
- initial velocity
- m/s
- a
- acceleration
- m/s²
- t
- time
- s
Velocity without time
v² = u² + 2as
When time is not given. The solver returns the positive root.
- v
- final speed
- m/s
- u
- initial speed
- m/s
- a
- acceleration
- m/s²
- s
- displacement
- m
Worked examples
Average speed of a cyclist
A cyclist travels 240 m along a straight track in 30 s. What is her average speed?
- Distance
- 240 m
- Time
- 30 s
Try it on paper first. Then open the working one step at a time.
A car pulling away
A car starts from rest and reaches 20 m/s in 8.0 s at constant acceleration. Find the acceleration and the distance travelled.
- u
- 0 m/s
- v
- 20 m/s
- t
- 8.0 s
Try it on paper first. Then open the working one step at a time.
Checks
A train travels 54 km in 45 minutes. What is its average speed in m/s? Convert both units before you divide.
A motorcycle starts from rest and accelerates at 3.0 m/s² for 6.0 s. How far does it travel, in metres?
A ball is thrown straight up at 15 m/s. Take g = 10 m/s² and upward as positive. How high does it rise, in metres, to 2 decimal places?
A velocity–time graph is a horizontal line above the time axis. The motion is
Easy places to lose a mark
- Mixing km/h with seconds. Divide km/h by 3.6 to get m/s.
- Using distance in the acceleration equations when the object has changed direction. Those equations want displacement.
- Forgetting that “from rest” means u = 0, not v = 0.
Practical
Motion track
Check the constant-acceleration equations against a timed run.