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