CIE AS Maths: Mechanics

Revision Notes

2.2.1 Using Calculus in 1D

Test Yourself

Using Calculus in 1D

How is calculus used in kinematics?

2.2.1 Using Calculus in 1D Diagram 1, downloadable Edexcel A Level Mechanics revision notes

  • s, v and a are all functions of time
  • 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
  • The use of calculus can be seen using velocity-time graphs
    • Acceleration is the gradient which is found using differentiation
    • Displacement is the area under the graph which is found using integration

    2-2-1-using-calculus-in-1d-diagram-2

How do I find c (the constant of integration) in kinematics?

  • The value of c can be found by substituting known values, as in integration for pure maths
  • Information to find c may be given in words – look out for common phrases …
    • “… starting from rest …”: this means v = 0 when t = 0
    • “… initially …” – this means t = 0

Definite integration in kinematics

  • Integrating acceleration between two values of time gives the change in velocity of the object between those times
  • Integrating velocity between two values of time gives the displacement of the object between those times
    • This could be different to the distance travelled between those times
    • To find the total distance travelled you would need to calculate the areas above and below the x-axis separately

Worked example

2.2.1_WE_Using Calculus in 1D_1

(a)  Find the acceleration of the particle in terms of t.

2-2-1-using-calculus-in-1d-example-solution-1

2-2-1-using-calculus-in-1d-example-solution-a

(b)  Find the displacement of the particle, from its starting position, after 3 seconds.

2-2-1-using-calculus-in-1d-example-solution-b

Exam Tip

  • Look for key words and phrases that indicate scalar, rather than vector quantities:
    • “distance …”, “speed” and “magnitude of …” are scalar
    • Displacement, velocity, acceleration are vectors
  • A sketch of the velocity-time graph is always helpful as you can use it to see important features such as times when the object is stationary. This is especially useful you are asked to find the total distance travelled.

You've read 0 of your 0 free revision notes

Get unlimited access

to absolutely everything:

  • Downloadable PDFs
  • Unlimited Revision Notes
  • Topic Questions
  • Past Papers
  • Model Answers
  • Videos (Maths and Science)

Join the 80,663 Students that ❤️ Save My Exams

the (exam) results speak for themselves:

Did this page help you?

Dan

Author: Dan

Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.