Displacement / Time Graphs

Here is a simple displacement time graph for someone walking at a steady speed.

The velocity is constant and is clearly equal to the gradient of the line. v = Δs / Δt

But what about this graph of displacement against time?

On this graph the object is speeding up. Its velocity, and hence the gradient of the graph, is increasing. If we divide the total displacement by time this will tell us the average velocity. The shape of the graph doesn't matter.

What if we want to know the velocity at any point in time, e.g. at t = 4 seconds?

So the velocity at any point in time is equal to the gradient of the graph at that point, i.e. the gradient of the tangent at that point.

We call this the "rate of change" of displacement and write it as       v = ds / dt