Distance-time graphs

Drawing a journey

Write down how far a cyclist has gone every few seconds and plot the points: time across, distance up. The line through them is a distance-time graph. It tells the whole story of the journey at a glance.

Reading the shapes

• A flat line: the distance does not change, so the object is at rest.
• A straight sloping line: equal distances in equal times, a steady speed.
• A steeper line: more distance each second, a higher speed.
• A line that curves upward: the speed is growing.

The slope is the speed

Between two points on a straight line,

You should find that B's steeper line crosses A's at the moment B catches up. If B is slower, B's line is less steep and never crosses.

Watch out. A distance-time graph is not a map of the road. A line going up does not mean going uphill; it means going further from the start.

A worked example

A cyclist's distance is written down every $10\ \text{s}$: $0, 40, 80, 80, 140, 200\ \text{m}$. Describe the trip.

1. First $20\ \text{s}$. $40\ \text{m}$ every $10\ \text{s}$: a steady $4\ \text{m/s}$.
2. $20\ \text{s}$ to $30\ \text{s}$. No change: she stopped at a signal.
3. Last $20\ \text{s}$. $60\ \text{m}$ every $10\ \text{s}$: $6\ \text{m/s}$, faster than before.
4. Average speed. $\dfrac{200\ \text{m}}{50\ \text{s}} = 4\ \text{m/s}$, counting the stop.

Practise with new numbers

Try it in Python

The program finds the speed in each gap of the example.

Remember these

• Flat line: at rest. Straight sloping line: steady speed. Curving up: speeding up.
• The slope of a distance-time graph is the speed.
• Average speed counts the stops in the total time.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. What does a flat part of a distance-time graph mean?

2. Two lines start at the origin. Line P is steeper than line Q. Which object is faster?

3. A straight line goes from $(0, 0)$ to $(25\ \text{s}, 150\ \text{m})$. What is the speed?

4. Can a distance-time graph ever slope downward?