Sketching Travel Graphs (Cambridge (CIE) O Level Additional Maths): Revision Note

Sketching travel graphs

How are s-t, v-t, and a-t graphs related?

• Recall that:

  • Velocity, v, is the rate of change of displacement, s, with respect to time

  • Acceleration, a, is the rate of change of velocity, v, with respect to time

  • Differentiate to go from s to v and from v to a

  • Integrate to go from a to v and from v to s

    • There will be a constant of integration, c, each time you integrate

• On a velocity-time graph:

  • Acceleration is the gradient which is found using differentiation

  • Displacement is the area under the graph which is found using integration

• This can also be seen from the units:

  • $\mathrm{Gradient}=\frac{{\mathrm{ms}}^{-1}}{\mathrm{s}}={\mathrm{ms}}^{-2}=\mathrm{Acceleration}$

  • $\mathrm{Area}={\mathrm{ms}}^{-1}\times \mathrm{s}=\mathrm{m}=\mathrm{Displacement}$

• On a displacement-time graph:

  • Velocity is the gradient which is found using differentiation

  • The area has no significant meaning

• This can also be seen from the units:

  • $\mathrm{Gradient}=\frac{\mathrm{m}}{\mathrm{s}}={\mathrm{ms}}^{-1}=\mathrm{Velocity}$

• On an acceleration-time graph:

  • Velocity is the area under the graph which is found using integration

  • The gradient is generally not used

    • It is a measure called 'jolt' but this is beyond the scope of this course

• This can also be seen from the units:

  • $\mathrm{Area}={\mathrm{ms}}^{-2}\times \mathrm{s}={\mathrm{ms}}^{-1}=\mathrm{Velocity}$

How can I use one travel graph to draw another?

• Using the relations stated above, we can inspect either the graph or the equation of the graph, in order to sketch a related travel graph

• For example, if a velocity-time graph is a series of sections, with mostly straight lines:

  • Find the gradient of each section to plot the acceleration-time graph

    • Remember if the gradient is negative, the acceleration-time graph will be below the x-axis

  • Find the area underneath each section to help plot the displacement-time graph

    • Remember that if the velocity is a positive constant (a horizontal line above the x-axis), the displacement will be increasing (a line with positive gradient)

    • If the velocity is a negative constant (a horizontal line below the x-axis), the displacement will be decreasing (a line with negative gradient)

• If a graph is a curve with a known equation, we can use calculus to find the equations of the other related functions

  • Remember you may need extra information about the velocity or displacement at a point in time when integrating

  • Once the equations of the other functions are found, they can be sketched

  • For example if the graph of the displacement-time graph is a cubic

    • the velocity-time graph will be a quadratic graph

    • and the acceleration-time graph will be a linear graph

Corresponding s-t, v-t, and a-t graphs for the same journey