Graphs of Motion

Physics: AQA A Level, topic 4.1.8

Displacement-Time Graphs

Displacement-time graphs have displacement on the y-axis and time on the x-axis.

Gradient

Gradient

If the object is not accelerating (travelling at a constant velocity), then the graph will be a straight line.
The gradient of a graph is defined as the change in y divided by the change in x.
On a displacement-time graph, y is the displacement and x is the time, so the gradient is the velocity:
Gradient $=\frac{\Delta y}{\Delta x} = \frac{\Delta s}{\Delta t} = v$

Uniform acceleration

Uniform acceleration

An object under uniform (constant) acceleration will have a displacement-time graph that curves upwards.
This is because the velocity (and therefore the gradient) is increasing.
The larger the acceleration, the faster the graph curves upwards.
If the object is decelerating, the velocity increases more slowly, so the graph becomes less steep.

Stationary

Stationary

The graph of a stationary object is a horizontal line.
The gradient of the graph is equal to zero as the average velocity of the object is zero.

Instantaneous and average velocity

Instantaneous and average velocity

To find the instantaneous velocity, draw a tangent to the curve at the point of interest and calculate its gradient.
The gradient of the tangent is the gradient of the curve at that point and therefore the velocity at that point.
To find the average velocity, simply divide the total change in displacement by the total time.

Instantaneous and average velocity

Instantaneous and average velocity

Velocity-Time Graphs

Velocity-time graphs have velocity on the y axis and time on the x axis.

Gradient

Gradient

The gradient of a graph is defined as the change in y divided by the change in x.
On a velocity-time graph, y is the velocity and x is the time, so the gradient is the acceleration:
gradient $=\frac{\Delta y}{\Delta x} = \frac{\Delta v}{\Delta t} = a$

Area under the graph

Area under the graph

The area under a velocity-time graph is the displacement of the object.

Non-uniform acceleration

Non-uniform acceleration

The graph above shows what a velocity-time graph looks like for non-uniform acceleration.

Acceleration-Time Graphs

Acceleration-time graphs have acceleration on the y-axis and time on the x-axis.

Uniform acceleration

Uniform acceleration

If the graph is positive, the object is accelerating.
If the graph is negative, the object is decelerating.
If the graph is at zero, there is no acceleration.
This means that the object is stationary or moving with constant velocity.

Area under the graph

Area under the graph

The area under an acceleration-time graph gives the object's change in velocity.

Non-uniform acceleration

Non-uniform acceleration

Any line on an acceleration-time graph that is not horizontal indicates a non-uniform acceleration.
This acceleration can be increasing or decreasing.
In this case, the object's acceleration decreases towards the ends of its motion.
Note that this is not the same as the object decelerating.

Test yourself on Graphs of Motion

If an object has constant positive acceleration what will the distance-time graph look like?
What will a velocity-time graph with constant velocity look like?
What is the area under an acceleration-time graph?
Which of these statements are true about graphs of motion?

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