Position - time graph and velocity

Accumulations set the connection between graphs

Physics is all about noticing, and then reusing, common patterns. You got a handle on velocity–time graphs by thinking about acceleration as the quantity which told velocity how to accumulate. By now, we hope you will not be surprised when we suggest getting a handle on displacement–time graphs by thinking about velocity as the quantity which tells displacement how to accumulate.

First consider discrete accumulations, to see the patterns in the two graphs.

Accumulating over smaller steps

One way of presenting a record of displacements is to use a scatter graph. You might consider several situations:

• The displacement is increasing (a positive velocity)  →  one characteristic shape.
• The displacement is not changing (the velocity is zero)  →  a different shape.
• The displacement is decreasing (a negative velocity)  →  yet another shape.

The necessary connection between velocity and displacement results in necessary connections between the shapes of the graphs representing a record of these values over time. So, you can move easily between the representations of velocity–time and displacement–time, so long as you remember that the connection between displacement and velocity is that velocity tells displacement how to change.

Here is a spelt-out summary of the connections:

• The greater the velocity, the greater the gradient of the displacement–time graph.
• If the velocity is positive, then the gradient will also be positive (sloping upwards as time increases).
• If the velocity is negative, then the gradient will also be negative (sloping downwards as time increases).
• If the velocity is zero then the gradient will also be zero (no slope at all).

Accumulating whilst the velocity changes

For a constant velocity the displacement increases by a fixed amount in each unit of time, and this varies as the value of the velocity changes. Therefore as the velocity varies, so the gradient of the displacement–time graph will vary. This gradient will be a curve. The curve can be concave or convex: it'll be set by the variations in the velocities. You might take a moment to figure out just why the curves have the shapes they do in the graphs on this page.

If the velocity is increasing then the increments will increase – a concave curve.

If the velocity is decreasing then the increments will get smaller – a convex curve.