Edexcel IGCSE Maths

Revision Notes

3.11.4 Differentiation - Kinematics

What is kinematics?

  • Kinematics is the analysis of the motion of a particle linking the three vector quantities displacement, velocity and acceleration – see below
  • Motion is in a straight line – think of the particle as moving along a number line
    • The number line has a fixed point O (the origin)
    • The number line has both negative and positive values
    • The particle can move in both directions along the number line
  • Note that in kinematics, a particle is an object – it could be a football, a car, a train – anything that has motion.  A particle is modelled as taking up a single point in space

 

Kin Notes fig1, downloadable IGCSE & GCSE Maths revision notes

 

  • Ensure you are familiar with Differentiation – Basics and
    Differentiation – Turning Points before continuing
  • It may be wise to look at Differentiation – Problem Solving too

What is displacement; isn’t it the same as distance?

  • Displacement is a vector quantity, so it can be negative
    • Distance is always positive
  • Displacement is measured from the fixed point O
  • The letter s is used for displacement
    • It is usually measured in metres (m)
  • If s = 4 then the distance from the origin is 4 m and the particle is 4 m “in front of” the origin
  • If s =-5 then the distance from the origin is 5 m and the particle is 5 m “behind” the origin
  • The + or – indicates the particle’s position relative to the origin

 

Kin Notes fig2, downloadable IGCSE & GCSE Maths revision notes

  • Displacement is a function of time, t, where time is usually measured in seconds
    • eg. s = 3t3 – 2t + 1
      At time t = 0, s = 1
      At time t = 2, s = 21

 

Kin Notes fig3, downloadable IGCSE & GCSE Maths revision notes

What is velocity; isn’t it the same as speed?

  • Velocity is a vector quantity, so it can be negative
    • Speed is always positive
  • The letter v is used for velocity
    • It is usually measured in metres per second (m/s)
  • If v = 3 then the speed of the particle is 3 m/s and it is moving in the positive direction
  • If v = -6 then the speed of the particle is 5 m/s and it is moving in the negative direction
  • The + or – indicates the particle’s direction of motion

 

Kin Notes fig4, downloadable IGCSE & GCSE Maths revision notes

  • Velocity is a function of time, t, and is the rate of change of displacement
    • To find v, differentiate s, ie. v = ds/dt
      If s = t3 – 2t2
      then v = ds/dt = 3t2 – 4t
  • If velocity is zero then the particle is stationary (not moving)

 

Kin Notes fig5, downloadable IGCSE & GCSE Maths revision notes

What is acceleration?

  • Acceleration is a vector quantity, so it can be negative
    • The magnitude of acceleration is always positive
  • The letter a is used for acceleration
    • It is usually measured in metres per square second (m/s2)
  • If a = 1 then the magnitude of acceleration is 1 m/s2 and the particle is accelerating (velocity increasing)
  • If a = -6 then the magnitude of acceleration is 6 m/s2 and the particle is decelerating (velocity decreasing)
  • The + or – indicates whether the particle is accelerating or decelerating

 

Kin Notes fig6, downloadable IGCSE & GCSE Maths revision notes

  • Acceleration is a function of time, t, and is the rate of change of velocity
    • To find a, differentiate v, ie. a = dv/dt
      If v = 3t2 – 4t
      then a = dv/dt = 6t – 4
  • If acceleration is zero then the particle is moving at a constant velocity

 

Kin Notes fig7, downloadable IGCSE & GCSE Maths revision notes

How do I solve kinematics problems?

  • Be clear about how the three quantities are related through differentiation
    • v = ds/dt
    • a = dv/dt

 

Kin Notes fig8, downloadable IGCSE & GCSE Maths revision notes

  • There are some key phrases to look out for
    • “… initial …” / “… initially …”
      This means at the start, so when t = 0
    • “… at rest …”
      This means the particle is stationary
      so v = 0
    • “… instantaneously …”
      This means at some point in time, for some value of t
  • For example,
    “Find the value(s)s of t for which the particle is instantaneously at rest”

    • means find the time(s) when v = 0,
    • ie. solve the equation v = 0

Kin Notes fig9, downloadable IGCSE & GCSE Maths revision notes

Exam Tip

Displacement, velocity and acceleration can all be negative whereas distance, speed and magnitude of acceleration are always positive.

Worked Example

Kin Example fig1 sol, downloadable IGCSE & GCSE Maths revision notes

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