Gradient of a Velocity-Time Graph

Acceleration is any change in the velocity of an object in a given time

$a c c e l e r a t i o n = \frac{c h a n g e i n v e l o c i t y}{t i m e} = \frac{(v - u)}{t} = \frac{∆ v}{∆ t}$

Where:
- v = final velocity (m s^–1)
- u = initial velocity (m s^–1)
- Δv = change in velocity (m s^–1)
- Δt = change in time (s)

As velocity is a vector quantity, this means that if the speed of an object changes, or its direction changes, then it is accelerating
- An object that slows down tends to be described as ‘decelerating’
The gradient of a velocity-time graph is equal to the acceleration

Acceleration on a velocity-time graph

velocity-time-gradient

The larger the gradient, the larger the acceleration represented on a velocity-time graph

Worked example

What does the velocity-time graph look like for this acceleration-time graph? WE - V-T gradient question image, downloadable AS & A Level Physics revision notes

Answer:

Step 1: Consider how velocity is represented on the acceleration-time graph

Acceleration is the gradient of a velocity-time graph

Step 2: Consider the velocity at each section of the acceleration-time graph

When the acceleration increases, the gradient of the velocity-time graph increases
When acceleration reaches a maximum, this is the maximum gradient of the velocity-time graph
When the acceleration decreases, the gradient of the velocity-time graph decreases

Exam Tip

A summary of the areas under the graph and gradients for the different motion graphs are:

Gradient:

The gradient of a displacement-time graph is the velocity
The gradient of a velocity-time graph is the acceleration

Area under the graph:

The area under a velocity-time graph is the displacement
The area under an acceleration-time graph is the velocity

Gradient of a Velocity-Time Graph (CIE AS Physics)

Revision Note