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When a relationship between two variables is defined by a curve it means that the gradient, or rate of change, is always varying.

An average speed for a journey can be found from a distance-time graph by working out the gradient of the line between the two points of interest. This is like using a car’s milometer and clock to take readings at the start and end of a journey.

However, they do not tell you the actual speed at a particular moment. You need a speedometer for that. On a graph, this can be estimated by drawing a tangent to the curve and calculating its gradient.

Example

This distance-time graph shows the first ten seconds of motion for a car.

To find an estimate of the speed after 6.5 seconds, draw the tangent to the curve at 6.5.

A velocity-time graph shows the velocity of a moving object on the vertical axis and time on the horizontal axis.

The gradient of a velocity time graph represents acceleration, which is the rate of change of velocity. If the velocity-time graph is curved, the acceleration can be found by calculating the gradient of a tangent to the curve.

Example

The velocity of a sledge as it slides down a hill is shown in the graph.

Find the acceleration of the sledge when t = 6s.

Notice, that after about 10 seconds, the gradients are negative meaning the sledge is slowing down or decelerating.