displacement-time graph slope

Displacement-time Graph Slope

The slope, or gradient, of a distance-time graph can tell us the velocity of a body.

Suppose a body moves at constant velocity. Suppose that at time t = 0 s, the displacement d is 0 m. A graph of the displacement against time would be a straight line through the origin. Suppose that after 5 s, the displacement is 2 m. Then the velocity v = d/t = 2/5 = 0.4 m/s. This also happens to be the slope, or gradient of the line. We can find the gradient by taking the line as one side of a triangle, and drawing in the vertical and horizontal lines to form the other sides. Then the gradient = vertical side / horizontal side. In this case, gradient = 2 / 5 = 0.4 m/s, the same as the velocity.

What if the graph is a curve? The velocity must be changing. Suppose we want to find the velocity at t = 3 s. First, locate the point in the graph. Then draw a straight line touching the curve, and find the gradient of this line. To find the gradient, draw a right angled triangle with the line as one side, the horizontal and vertical as the other two sides. Suppose the horizontal side is 2 s, and the vertical side is 0.5 m. Then the gradient = 0.5 / 2 = 0.25 m/s.

To see why this is so, think of a very small time interval about t = 3 s. Suppose this time interval is b seconds. Draw the two vertical lines to the curve that contains this time interval. At the curve, draw a horizontal line in between them to show the change in displacement, a metres. Notice that the sides a, b and the curve roughly forms a right angled triangle. So the velocity during this time is approximately given by the change in displacement divided by the time, i.e. a/b m/s. This is not exact because the curved side of the triangle is not straight.

Now imagine making the time interval b very small. The triangle becomes very small, the curved side becomes very short, and also much straighter. So a/b m/s would give the velocity much more accurately. It is difficult to measure such a tiny triangle. The easy way is to draw a longer, straight line touching the curve at t = 3 s. This line would be parallel to the slanted side of the triangle, so that its gradient is also equal to a/b.

Therefore the gradient, or slope, of the displacement time graph is equal to the velocity.

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