Motion Graphs

There are several types of graphs that can be used to analyse motion:

  • position-time graphs
  • velocity-time graphs
  • acceleration-time graphs

Qualitative and quantitative information about the motion of a body can be determined by considering the shape, gradient and area under various graphs. Understanding how to correctly identify and calculate these values for each different type of graph is the focus of this section.

Position-Time Graphs

A position-time graph will show the position, x, of an object measured from a starting position over a time interval, t. Position is on the vertical axis and has the SI unit metres (m) and time is on the horizontal axis with the SI unit seconds (s). The following information can be determined from a position-time graph:

  • The position (also displacement) of the object from its origin at any time by reading directly from the graph
  • If the object is moving toward or away from its original position.
  • The total distance covered by the object by adding up different sections of the objects position
  • The velocity of the object at any instant by calculating the gradient at that point

Gradient and Velocity

The velocity of an object at any time, t, is determined by calculating the gradient of the graph at that point. This is relatively straight forward for graphs of uniform motion – where an object is either stationary or moving at constant velocity, but, is more difficult for graphs of non-uniform motion (see below). The gradient is determined by calculating the \cfrac { rise }{ run } at any particular point:

Non-uniform Motion

Non-uniform motion is motion where the velocity is constantly changing. This is seen as a curve on a position-time graph.

Velocity-Time Graphs

A velocity-time graph will show the velocity, v, of an object over a time interval, t. Velocity is on the vertical axis and has the SI unit m/s, and time is on the horizontal axis with the SI unit seconds (s). The following information can be determined from a velocity-time graph:

  • The velocity of the object at any time by reading directly from the graph
  • If the object is speeding up or slowing down.
  • The total distance covered by the object by calculating the area under the graph
  • The acceleration of the object at any instant by calculating the gradient at that point

Gradient and Acceleration

The acceleration of an object at any time, t, is determined by calculating the gradient of the graph at that point. How to calculate the gradient was demonstrated above.

Negative Velocity

Some graphs may indicate a negative velocity as shown below. From the graph below it can be observed that:

  • the object is accelerating between 0-20 seconds
  • the object maintains a constant velocity of 8 m/s between 20-30 seconds
  • the object slows down between 30-50 seconds
  • the object increases its speed in the opposite direction between 50-65 seconds, and so on

*note: even though the object may be moving in the opposite direction as indicated when the line trends below the line – this does not necessarily mean the object is back past its origin as would be the case with a position-time graph.

Acceleration-Time Graphs

An acceleration-time graph will show the acceleration, a, of an object over a time interval, t. Acceleration is on the vertical axis and has the SI unit m/s2, and time is on the horizontal axis with the SI unit seconds (s). The following information can be determined from an acceleration-time graph:

  • The acceleration of the object at any time by reading directly from the graph
  • If the object is speeding up or slowing down.
  • The objects change in velocity by calculating the area under the graph

The acceleration-time graph below matches the velocity-time graph illustrated above.

Example 1:

Using the position-time graph below, complete the following questions:

  1. how far has the object travelled after 25 and 70 seconds?
  2. when is the object at rest?
  3. what is the velocity of the object between 0-25 seconds?
  4. what is the total displacement and distance of the object after 100 seconds?

Answers:

a) 10m, 40m (read directly from the graph)

b) between 25-40 and 70-80 seconds

c) velocity = gradient of the graph:

= \cfrac { 10-0 }{ 25-0 } = 0.4 m/s

d) displacement = 0m (graph finishes at origin), distance = 80m

Example 2:

Using the velocity-time graph below, complete the following questions:

  1. what is the velocity of the object after 10 and 70 seconds?
  2. when is the object moving at a constant velocity?
  3. when was the object stationary?
  4. what is the acceleration of the object between 0-20 seconds?
  5. what is the total displacement of the object after 50 seconds?

Answers:

a) 4 m/s, -6 m/s (read directly from the graph)

b) between 20-30 and 65-75 seconds

c) between 85-100 seconds

d) acceleration = gradient of the graph:

= \cfrac { 8-0 }{ 20-0 } = 0.4 m/s2

e) 240m (area under the graph between 0-50 seconds

Example 3:

Draw an acceleration-time graph for the following velocity-time graph:

Answer: