# 3.03 Velocity time graphs

The first diagram shows a velocity time graph for a car travelling with constant acceleration.
(In this example we are only considering motion in one dimension with no change in direction so the change in velocity in this case would only be caused by a change in speed.) The graph shows how the car's velocity increases over regular time intervals starting with an initial velocity of 0.

The gradient of the graph is equal to the cars acceleration i.e. a = Δv/Δt
As the car is travelling with constant acceleration the gradient of the graph is the same at all points,(i.e it has a constant gradient and therefore produces a straight line graph).

The graph below shows an example where the initial velocity is not zero.
Here the gradient of the graph would be (v-u)/t.
Therefore a = (v -u)/t

We will consider only very limited cases where acceleration changes. One case is where there is simply a different value of acceleration at different stages of an object's motion. In this case we just deal with one stage at a time (and the acceleration is constant during each stage). The first graph below shows one stage of a cars motion where it has a constant positive acceleration and then a second stage where its acceleration is zero (as can be seen from the gradients of the graph).
Another case we will consider is when the acceleration is changing constantly producing a curved graph as shown in the second diagram below. (This will be covered later when we examine drag forces).

## Area under a velocity time graph

The area under a velocity time graph is equal to the change in displacement Δs.

(In our examples we are taking the initial displacement to be zero so the change in displacement and the final displacement are the same ie Δs = s)