Motion & Speed
Distance, Speed, and Time
Speed is a measure of how far an object travels per unit of time.
Formula: Speed = Distance / Time
Rearranged:
- Distance = Speed x Time
- Time = Distance / Speed
Units: metres per second (m/s), kilometres per hour (km/h)
To convert km/h to m/s: divide by 3.6 To convert m/s to km/h: multiply by 3.6
Worked Example
A car travels 150 km in 2.5 hours. Calculate its average speed.
Speed = Distance / Time = 150 / 2.5 = 60 km/h
Now find how far it travels in 3 hours at this speed: Distance = Speed x Time = 60 x 3 = 180 km
Speed vs Velocity
| Speed | Velocity | |
|---|---|---|
| Type | Scalar | Vector |
| What it includes | Magnitude only | Magnitude AND direction |
| Example | 60 km/h | 60 km/h north |
A scalar quantity has magnitude only. A vector quantity has both magnitude and direction.
An object moving in a circle at constant speed is changing velocity because its direction is constantly changing. This means it is accelerating even if its speed does not change.
Other scalar quantities: distance, mass, temperature, energy Other vector quantities: displacement, force, acceleration, momentum
Distance-Time Graphs
Distance-time graphs show how far an object has moved over time.
| Feature on graph | What it means |
|---|---|
| Horizontal line | Stationary (not moving) |
| Straight diagonal line upward | Constant speed |
| Steeper gradient | Greater speed |
| Curved line (increasing gradient) | Accelerating |
| Curved line (decreasing gradient) | Decelerating |
| Line going back down | Returning to start |
Key rule: Gradient of a distance-time graph = speed
To calculate speed from the graph: pick two points on the line and calculate rise / run (change in distance / change in time).
Speed-Time Graphs
Speed-time graphs show how the speed of an object changes over time.
| Feature on graph | What it means |
|---|---|
| Horizontal line | Constant speed |
| Upward sloping line | Acceleration |
| Downward sloping line | Deceleration |
| Steeper slope | Greater acceleration |
Key rules:
- Gradient of a speed-time graph = acceleration
- Area under a speed-time graph = distance travelled
To find the area under a speed-time graph: calculate the area of rectangles and triangles formed.
Acceleration
Acceleration is the rate of change of velocity.
Formula: a = (v - u) / t
Where:
- a = acceleration (m/s squared)
- v = final velocity (m/s)
- u = initial velocity (m/s)
- t = time (s)
A negative acceleration means the object is slowing down — this is called deceleration.
Worked Example — Acceleration
A cyclist accelerates from rest (0 m/s) to 8 m/s in 4 seconds. Find the acceleration.
a = (v - u) / t = (8 - 0) / 4 = 2 m/s squared
Now find the distance covered (assuming constant acceleration — use area of triangle): Distance = 0.5 x base x height = 0.5 x 4 x 8 = 16 m
Typical Speeds
It is useful to know approximate speeds for common situations:
| Situation | Approximate speed |
|---|---|
| Walking | 1.5 m/s |
| Running | 3 m/s |
| Cycling | 6 m/s |
| Car on motorway | 30 m/s |
| Sound in air | 340 m/s |
| Light in vacuum | 300,000,000 m/s |
Key Terms
- Speed — distance travelled per unit time (scalar, m/s)
- Velocity — speed in a given direction (vector, m/s)
- Acceleration — rate of change of velocity (m/s squared)
- Deceleration — negative acceleration; slowing down
- Scalar — quantity with magnitude only
- Vector — quantity with magnitude and direction
- Gradient — steepness of a line on a graph (rise / run)
Common Mistakes
- Confusing distance (scalar — how far travelled) with displacement (vector — straight-line distance from start to finish)
- Reading a horizontal line on a distance-time graph as constant speed — it means stationary
- Forgetting that area under a speed-time graph gives distance, not displacement
- Using the wrong formula — write down s = d/t, then rearrange before substituting numbers
Tips and Tricks
- Use the formula triangle: write D on top, S and T on the bottom — cover the unknown to get the formula
- On a distance-time graph: flat = stopped, slope = moving; steeper = faster
- On a speed-time graph: flat = constant speed, slope = accelerating/decelerating
- Remember: area under speed-time graph = distance (count squares or use geometry)
- Deceleration is just acceleration with a negative value — do not treat it as a separate formula