Calculus for Kinematics (DP IB Applications & Interpretation (AI)): Revision Note

Differentiation for Kinematics

How is differentiation used in kinematics?

• Displacement, velocity and acceleration are related by calculus

• In terms of differentiation and derivatives

  • velocity is the rate of change of displacement

    •   or  

  • acceleration is the rate of change of velocity

    •   or  

  • so acceleration is also the second derivative of displacement

    •   or  

• If a graph is not given you can use your GDC to draw one

  • you can then use your GDC’s graphing features to find gradients

    • velocity is the gradient on a displacement (-time) graph

    • acceleration is the gradient on a velocity (-time) graph

Integration for Kinematics

 How is integration used in kinematics?

• Since velocity is the derivative of displacement () it follows that

• Similarly, velocity will be an antiderivative of acceleration

How would I find the constant of integration in kinematics problems?

• A boundary or initial condition would need to be known

  • phrases involving the word “initial”, or “initially” are referring to time being zero, i.e. 

  • you might also be given information about the object at some other time (this is called a boundary condition)

  • substituting the values in from the initial or boundary condition would allow the constant of integration to be found

How are definite integrals used in kinematics?

• Definite integrals can be used to find the displacement of a particle between two points in time

  •  would give the displacement of the particle between the times and

    • This can be found using a velocity-time graph by subtracting the total area below the horizontal axis from the total area above

  •  gives the distance a particle has travelled between the times and

    • This can be found using a velocity velocity-time graph by adding the total area below the horizontal axis to the total area above

    • Use a GDC to plot the modulus graph