Displacement-Time Graphs (Cambridge International A-Level Mathematics)

Displacement-time graphs

What is a displacement-time graph?

• Displacement-time graphs show the displacement of an object from a fixed origin as it moves in a straight line

• They show displacement(on the vertical axis) against time (on the horizontal axis)

• Displacement-time graphs can go below the horizontal axis whereas distance-time graphs can not

  • Distance can not be negative whereas displacement can be

What are the key features of a displacement-time graph?

• The gradient of the graph equals the *velocity of the object

  • A positive gradient means the object is travelling forwards
  • A negative gradient means that the object is travelling backwards
  • The steeper the line, the greater the speed

• A straight line shows that the object is moving at a constant velocity

• A curved line shows that the object is accelerating or decelerating

• A horizontal line shows that the object is stationary

• If the graph touches the x-axis, then the object is at the origin at that time

How do I find the average speed and the average velocity of a journey?

$\mathrm{Average} \mathrm{speed}=\frac{\mathrm{Total} \mathrm{distance} \mathrm{travelled}}{\mathrm{Time} \mathrm{taken}}$

• The average speed can not be negative as speed is a scalar quantity and can only take a positive value

$\mathrm{Average} \mathrm{velocity}=\frac{\mathrm{Displacement} \mathrm{from} \mathrm{starting} \mathrm{point}}{\mathrm{Time} \mathrm{taken}}$

• The average velocity is a vector quantity and can be positive, zero or negative

Worked Example An athlete is training by jogging in a straight line.

The displacement-time graph below shows the displacement of the athlete from her starting point throughout her motion.

(a) Calculate the velocity of the athlete during the first 2 seconds.

Answer:

(b) Describe the motion of the athlete between the times of 2 seconds and 5 seconds.

Answer:

(c) Calculate the velocity of the athlete at 10 seconds.

Answer:

(d) Find the total distance travelled by the athlete during the 14 seconds.

Answer:

Exam Hint

• Be careful to spot if you are working with a displacement-time graph or a velocity-time graph.
• Be careful to spot if you are working with a displacement-time graph or a distance-time graph
• Check where the graph starts from on the y-axis, the object does not have to start at 0. For example, if the graph starts at 100, the scenario could be a student on a walk starting 100 m from their house.
• Distance is a scalar so it can not be negative whereas displacement from a starting point can be.
• Think about the units when calculating, make sure they are consistent.