# Velocity from a Displacement-Time Graph – how to derive?

Velocity is the ratio of the displacement and the time duration. Hence, **velocity can be obtained from the gradient of a displacement-time graph.**

Velocity = gradient of displacement-time graph

**Drawing a displacement-time graph from a set of data**

Let’s take a case study of a toy car that moves along a straight track. Its displacement at different times is shown in the table below. Then we will use this data to draw a displacement-time graph from which we can deduce the car’s velocity.

Displacement s (m) | 1 | 3 | 5 | 7 | 7 | 7 |

Time t (s) | 0 | 1 | 2 | 3 | 4 | 5 |

**data**

**A set of**displacement-timeNow we can plot the displacement-time graph as the following one.

**Deriving velocity from a displacement-time graph**

We draw a right-angled triangle as shown. Now, to find the car’s velocity, we need to divide a displacement by a time. These are given by the two sides of the triangle labeled Δs and Δt.

Velocity v = change in displacement/change in time = Δs/Δt = (7-1)/(3-0) = 6/3 m/s = 2 m/s

**Summary**:

The gradient of a displacement-time graph gives velocity.

v = Δs/Δt