Calculus for Kinematics (DP IB Maths: AI HL): Revision Note

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  
  • Sometimes velocity may be a function of displacement rather than time
    •  rather than
    • in such circumstances, acceleration is
    • this result is derived from the chain rule
  • All acceleration formulae are given in the formula booklet
• Even if a motion graph is given, if possible, 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
• Dot notation is often used to indicate time derivatives
  •  is sometimes used as displacement (rather than) in such circumstances
  • , so  is velocity
  • , so  is acceleration

 How is integration used in kinematics?

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

• Similarly, velocity will be an antiderivative of acceleration

• You might be given the acceleration in terms of the velocity and/or the displacement
  • In this case you can solve a differential equation to find an expression for the velocity in terms of the displacement

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