Motion Graphs

The question

A sprinter's run is recorded as distance against time. When does she reach top speed, and what is it?

She reaches top speed at 4 s, 20 m into the run. That is where her displacement–time graph stops curving and becomes a straight line. Her top speed is the gradient of that straight part: 80 m in 8 s, which is 10 m/s.

In words, then in symbols

• Velocity is the change in displacement divided by the time it takes: the gradient of the displacement–time graph.

  $v={\displaystyle \frac{\mathrm{\Delta }s}{\mathrm{\Delta }t}}$

• On a velocity–time graph the rise is a change in velocity and the step along is a time, so the gradient is the acceleration.

  $a={\displaystyle \frac{\mathrm{\Delta }v}{\mathrm{\Delta }t}}$

How to read a motion graph

1. Read what is on each axis, with its units.
2. The gradient is how fast the quantity on the side axis changes: on a displacement–time graph that is the velocity, on a velocity–time graph the acceleration.
3. On a curve, draw a tangent and find its gradient.
4. On a velocity–time graph, the area under the line is the displacement.
5. Say the shape in words: flat, straight and sloping, or curving.