Speed, distance and time: formula, graphs and examples

Draw a triangle with d on top and s and t at the bottom. Cover the quantity you want: covering $d$ leaves $s \times t$; covering $s$ leaves $d \div t$; covering $t$ leaves $d \div s$. It's a memory aid; in exams, write the equation in full.

1. A cyclist rides 1,800 m in 5 minutes. Convert: 5 minutes = 300 s. Speed $= 1800 \div 300 = 6$ m/s.
2. A train travels at 50 m/s for 2 minutes. Time = 120 s, so distance $= 50 \times 120 = 6{,}000$ m (6 km).
3. How long does it take to walk 450 m at 1.5 m/s? Time $= 450 \div 1.5 = 300$ s, which is 5 minutes.

• A straight sloping line means constant speed. The steeper the line, the faster.
• A horizontal line means the object is stationary.
• A curve means the speed is changing.

The gradient (change in distance ÷ change in time) is the speed. For a curve, draw a tangent and find its gradient for the speed at that moment.

Acceleration is the change in velocity per second:

measured in m/s². On a velocity-time graph:

• the gradient is the acceleration (a negative gradient is deceleration);
• the area under the line is the distance travelled.

Example: a car speeds up from 0 to 20 m/s in 8 s. $a = 20 \div 8 = 2.5$ m/s². The area under that part of the graph is a triangle: $\tfrac{1}{2} \times 8 \times 20 = 80$ m.

Speed is a scalar: it has size only. Velocity is a vector: speed in a given direction. A car driving round a roundabout at a steady 10 m/s has a constant speed but a changing velocity, because its direction changes. That's why it's accelerating even though the speedometer doesn't move. More on forces and acceleration in Newton's laws of motion.

• Mixing units, such as km with seconds. Convert first.
• Forgetting that distance-time and velocity-time graphs mean different things: a flat line is stationary on one and constant speed on the other.
• Reading the area of a velocity-time graph incorrectly for triangles (remember the ½).

1. A student walks 2 km in 8 minutes. What is their average speed in m/s?
2. A car travels at 25 m/s for 40 s. How far does it go?
3. A car speeds up from 10 m/s to 30 m/s in 5 s. What is its acceleration?
4. How long does it take to run 400 m at 8 m/s?

Answers: about 4.17 m/s ($2000 \div 480$); 1,000 m; 4 m/s²; 50 s.