Edexcel IGCSE Maths 复习笔记 3.11.4 Differentiation - Kinematics

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 displacementvelocity 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

 

3.11.4-Kin-Notes-fig1

  • 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

 

3.11.4-Kin-Notes-fig2

 

 

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

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

     

 

3.11.4-Kin-Notes-fig3

 

 

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

 

3.11.4-Kin-Notes-fig4 

 

  • Velocity is a function of timet, and is the rate of change of displacement
    • To find vdifferentiate 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)

 

 3.11.4-Kin-Notes-fig5 

 

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

 

 3.11.4-Kin-Notes-fig6 

 

  • Acceleration is a function of timet, 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

 

 3.11.4-Kin-Notes-fig7 

 

How do I solve kinematics problems?

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

     

 

3.11.4-Kin-Notes-fig8

 

 

  • 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

     

 

 

3.11.4-Kin-Notes-fig9

Exam Tip

Displacementvelocity and acceleration can all be negative whereas distancespeed and magnitude of acceleration are always positive.

Worked Example

3.11.4-Kin-Example-fig1-sol

转载自savemyexam

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