Distance-Time and Speed-Time Graphs

This lesson explains Distance-Time and Speed-Time Graphs for Cambridge IGCSE Physics 0625. It separates the Core requirements from the additional Supplement work for Extended candidates. Pay close attention to equation choice, unit conversion and the final sense check. The goal is to explain the idea accurately and apply it in an unfamiliar context.

What does the gradient of each graph tell you?

On a distance-time graph, the gradient gives speed. On a speed-time graph, the gradient gives acceleration. Mixing these two readings is the single most common graph error in 0625.

Graph	Gradient means	Area under graph means	Horizontal line means
Distance-time	speed	nothing useful	object at rest
Speed-time	acceleration	distance travelled	constant speed

In words: speed equals the change in distance divided by the change in time, read from the slope. Acceleration equals the change in speed divided by the change in time, read from the slope of a speed-time graph. In symbols: $v = \dfrac{\Delta s}{\Delta t}$ and $a = \dfrac{\Delta v}{\Delta t}$.

How do you find distance from a speed-time graph?

Calculate the area under the line. For constant speed, that area is a rectangle: $\text{distance} = \text{speed} \times \text{time}$. For uniform acceleration from rest, it is a triangle: $\text{distance} = \dfrac{1}{2} \times \text{time} \times \text{final speed}$. For mixed journeys, split the shape into rectangles and triangles and add the areas. Extended students must also recognise curves: an increasing gradient on a speed-time graph means increasing acceleration, and a curve flattening towards horizontal means acceleration decreasing to zero, the terminal velocity shape.

Worked example

A cyclist starts from rest and accelerates uniformly to 6.0 m/s in 8.0 s. She then travels at a constant 6.0 m/s for 12 s. (a) Calculate her acceleration during the first 8.0 s. [2] (b) Calculate the total distance travelled in the 20 s journey. [3]

Solution (a). Equation: $a = \dfrac{\Delta v}{\Delta t} = 6.0 \div 8.0 = 0.75\ \text{m/s}^2$.

Solution (b). Distance = area under the speed-time graph. Triangle: $\dfrac{1}{2} \times 8.0 \times 6.0 = 24\ \text{m}$. Rectangle: $6.0 \times 12 = 72\ \text{m}$. Total: $24 + 72 = 96\ \text{m}$.

Original marking points

• M1 (a): $a = \dfrac{\Delta v}{\Delta t}$ or $6.0/8.0$ seen. A1: $0.75\ \text{m/s}^2$ with unit.
• M1 (b): $\dfrac{1}{2} \times 8.0 \times 6.0 = 24\ \text{m}$ (triangle area).
• M1 (b): $6.0 \times 12 = 72\ \text{m}$ (rectangle area).
• A1 (b): $96\ \text{m}$.

These marking points belong to this original example. They are not an official Cambridge mark scheme.

Common errors and how to correct them

• Reading a distance-time gradient as acceleration. Fix: check the y-axis label before you write anything.
• Forgetting the $\dfrac{1}{2}$ in the triangle area. That doubles the distance. Fix: sketch the shape and label it triangle or rectangle first.
• Saying a horizontal line on a speed-time graph means “stopped”. It means constant speed; stopped is the line along $v = 0$. Fix: learn both horizontal-line meanings from the table above.
• Using one point instead of a gradient triangle. Fix: take the change in y over the change in x across a large section of the line.
• Misreading scales. A small square is not always one unit. Fix: write the value of one square on the axis before calculating.

How to apply this in an exam

Before answering any graph question, annotate the graph itself: label each section “accelerating”, “constant speed” or “at rest”, and write the gradient or area you need next to it. Examiners mark working on the graph, and this 20-second habit prevents the axis mix-up that costs the whole question.

Where this skill matters

Graphs span Core and Extended. Papers 1 and 2 give a graph and ask which statement describes the motion, worth one mark with a high error rate. Papers 3 and 4 set structured questions like the worked example; Paper 4 (Extended) adds curved graphs, tangent gradients and terminal-velocity descriptions worth up to 6 marks. Paper 6 can ask you to plot motion data, draw a best-fit line and find a gradient, so bring a sharp pencil and a 30 cm ruler. Core candidates stick to straight-line graphs; Extended candidates must handle changing acceleration too.