4.1.8

Graphs of Motion

Test yourself

Displacement-Time Graphs

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

• 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

• 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

• 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

• 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.

Velocity-Time Graphs

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

• 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

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

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

• 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

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

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.