How is calculus used in kinematics?

• 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

  

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

   

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

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