The velocity-time graph

Distance travelled

Average velocity is defined by the equation of section 3

In general, averages may be found by adding the available values and dividing by the number of values available. So an alternative way of finding the average speed is to add the initial and final speeds, and then divide by 2:

Combining these two equations, we obtain

Here is a section of a speed-time graph

The area of the green trapezium is

area = ‘base’ ´ average ‘height’

(the quotes indicating that we take the scaled heights and base rather than the literal heights and base). It is easy to see, since the start and finish ‘heights’ are just the initial and final speeds, that the area formula and the distance travelled formula are identical. Thus

distance travelled = ‘area’ under a velocity-time graph

and an easy way to calculate this is by working out the average ‘height’ and then multiplying by the ‘base’.

Acceleration

Here is the same velocity-time graph. You will notice that the speed gets steadily bigger.

This time we work out the gradient. Referring to the gradient triangle, drawn in red

Since the last two expressions are the definition of acceleration, we conclude that

acceleration = gradient of a velocity-time graph.