Position-time graphs

A story in one picture

A cyclist rides to a friend's house, stops to chat, and rides back part of the way to buy a snack. You could describe this in words. A position-time graph tells it in one picture: position $x$ up the side, time $t$ along the bottom.

Four shapes to know

• A flat line: the position does not change. The body is at rest.
• A straight line going up: the same increase in $x$ every second. A steady positive velocity.
• A straight line going down: a steady negative velocity: the body moves towards smaller $x$.
• A curve: the slope changes, so the velocity changes. Getting steeper means speeding up; getting flatter means slowing down.

The rule behind all four: the slope of a position-time graph is the velocity.

Draw your own

Choose two velocities and the moment the body switches from one to the other. The red dot on the position axis is the body itself: watch it move up and down the axis as the graph says.

Reading a graph: a worked example

A cyclist's readings: at $0\ \text{s}$ she is at the start; at $20\ \text{s}$ she is $100\ \text{m}$ along; she waits there until $50\ \text{s}$; at $60\ \text{s}$ she is back at $40\ \text{m}$; and at $90\ \text{s}$ she is at $160\ \text{m}$.

1. $0\ \text{s}$ to $20\ \text{s}$. Slope $\dfrac{100\ \text{m}}{20\ \text{s}} = 5\ \text{m/s}$.
2. $20\ \text{s}$ to $50\ \text{s}$. Flat: at rest.
3. $50\ \text{s}$ to $60\ \text{s}$. Slope $\dfrac{40\ \text{m} - 100\ \text{m}}{10\ \text{s}} = -6\ \text{m/s}$: she rides back.
4. $60\ \text{s}$ to $90\ \text{s}$. Slope $\dfrac{160\ \text{m} - 40\ \text{m}}{30\ \text{s}} = 4\ \text{m/s}$.
5. Over the whole $90\ \text{s}$. Displacement $160\ \text{m}$, so the average velocity is $1.78\ \text{m/s}$. Path length $100\ \text{m} + 60\ \text{m} + 120\ \text{m} = 280\ \text{m}$, so the average speed is $3.11\ \text{m/s}$.

Watch out. The height of the graph is a position, not a speed. A high flat line is a body at rest far from the origin. Only the slope tells how fast.

Two bodies on one graph

Draw two bodies' lines on the same axes. Where the lines cross, the two are at the same place at the same time: one meets or overtakes the other. A steeper line is a faster body.

Think first. Two lines are parallel. What does that tell you? The two bodies have the same velocity, so the gap between them never changes, and they never meet.

Practise with new numbers

Try it in Python

The program draws the cyclist's graph from her readings and works out the slope of each straight piece.

Remember these

• Slope of a position-time graph = velocity. Height = position.
• Flat: at rest. Straight: steady velocity. Curve: changing velocity.
• Crossing lines: the two bodies meet. Parallel lines: they never do.

Check yourself

1. A position-time graph is a straight line through $(2\ \text{s}, 30\ \text{m})$ and $(8\ \text{s}, 6\ \text{m})$. What is the velocity?

2. Sketch in words the position-time graph of a ball rolled away from you that slows down and stops.

3. On one graph, line A is steeper than line B, and both rise. Which body is faster? Must A be ahead of B?

4. Can a position-time graph ever be a vertical line? Explain.