Reading Kinematics Graphs



Distance-Time Graphs

distance time graph
  • For a distance-time graph, the distance never decreases.
  • When the object is stationary, the distance-time graph will be horizontal.
  • The gradient of a distance-time graph is the instantaneous speed of the object.
  • For straight line with positive gradient, it means that the object is travelling at uniform speed
  • There is no straight line with negative gradient (as the distance never decreases)
  • For curves, it means that the object is travelling at non-uniform speed

Displacement-Time Graphs

  • The details are similar as distance-time graphs, except that the distance is now displacement, and speed is now velocity.
  • The gradient of a distance-time graph is the instantaneous velocity of the object.
  • One minor difference: There is a straight line with negative gradient, it means that the object is travelling at uniform velocity in the opposite direction.

Velocity-Time Graphs

velocity-time graph
  • When the object is stationary, it is a straight horizontal line at 0.
  • When the object is undergoing uniform motion, it is a straight horizontal line at $v \, \text{m s}^{-1}$, where v is the velocity of the object.
  • The gradient of a velocity-time graph of an object gives the acceleration of the object.
  • For straight line with positive gradient, it means that the object is accelerating.
  • For straight line with negative gradient, it means that the object is decelerating.
  • For curves, it means that the acceleration of the object is changing.
  • The area under the graph is the change in the displacement of the object.

Acceleration-Time Graphs

  • Area under graph is the change in the velocity of the object

Summary of Kinematics Graphs

The figure below shows the displacement-time graph, velocity-time graph and acceleration-time graph for the respective state of motion. It serves as a summary of the text above.

all graphs of motion

The figure below shows the relationship between displacement-time graph, velocity-time graph and acceleration-time graph.

Graphs of motion

Worked Examples

Example 1

Can you tell from a displacement-time graph whether an object is stationary?

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Yes. If the object is stationary, it will appear as a horizontal line on a displacement-time graph.

Example 2

How can you obtain the average velocity and instantaneous velocity from a displacement-time graph.

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The average velocity can be found by using $\frac{\text{total displacement}}{\text{total time taken}}$.

The instantaneous velocity at a point in time can be found from the gradient of the tangent to that point in time.

Example 3

Can you tell from a velocity-time graph whether an object is stationary?

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Yes. If the object is stationary, the velocity-time graph will be a horizontal line at $\text{v}=0$.

Example 4

How would you obtain the acceleration of an object from a velocity-time graph? What does the area under a velocity-time graph represent?

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The acceleration of an object at a point in time can be obtained from the gradient of the tangent to that point in time.

The area under a velocity-time graph represents the total distance traveled.

Example 5

Can you tell from an acceleration-time graph whether an object is stationary?

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No, you cannot. Do you know why?

Hint: Refer to the summary of Kinematics graphs (located above).

Drop a comment below if you cannot figure out the answer.

Example 6

The acceleration-time graph of an object moving in a straight line is as shown. The object started its motion from rest.

acceleration-time graph qn

At which point is the body moving with the largest speed?

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Area under acceleration-time graph = change in velocity of the object. At point 2, the area under the graph is the largest.

Answer: 2


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