Edexcel IGCSE Maths 复习笔记 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

• Ensure you are familiar with Differentiation - Basics andDifferentiation – 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

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

    At time t = 0, s = 1At time t = 2, s = 21

   

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

• 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 - 2t²then v = ds/dt = 3t² – 4t

   

• If velocity is zero then the particle is stationary (not moving)

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

• 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 – 4tthen a = dv/dt = 6t - 4

   

• If acceleration is zero then the particle is moving at a constant velocity

How do I solve kinematics problems?

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

   

• 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 stationaryso 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