Stopping Distance: Velocity and Acceleration

The question

If you drive twice as fast, how much further do you need to stop?

Twice the speed makes the thinking distance twice as long and the braking distance four times as long, because the speed appears twice in the braking triangle: in its height and in its width. From 10 m/s to 20 m/s the stopping distance goes from 17 m to 54 m, just over three times as far. The faster the car, the more of the stopping distance is braking, and the closer the total gets to four times as far.

In words, then in symbols

• At a steady speed, the distance moved is the speed multiplied by the time.

  $d=vt$

• Acceleration is how fast velocity changes: the change in velocity divided by the time the change takes.

  $a={\displaystyle \frac{v-u}{t}}$

• The stopping distance is the speed times the reaction time, plus the speed squared divided by twice the deceleration.

  $d=v{t}_r+{\displaystyle \frac{v^2}{2a}}$

How to find a distance from a velocity–time graph

1. Draw the velocity–time graph for the whole motion.
2. Split the area under the line into simple shapes: rectangles and triangles.
3. Find the area of each shape: a rectangle is base × height, a triangle is ½ × base × height.
4. Add the areas. The total is the distance travelled.