Using Differentiation for Kinematics (Edexcel IGCSE Maths B): Revision Note

Kinematics

What does kinematics mean?

• Kinematics is the study of the motion of an object

  • It is a branch of Physics

• Objects are called particles

  • They are modelled as single moving points

• Over time, the particles move and

  • can be at different distances from a fixed origin (displacement)

  • can move with different speeds in different directions (velocity)

  • can speed up or slow down (acceleration)

What is displacement?

• The displacement of an object is how far away it is from a fixed origin

  • It can be positive (in front of the origin)

  • or negative (behind the origin)

• Do not confuse displacement with distance

  • Distance is always positive!

  • Displacement can have a $\pm$ sign

• Displacement is given the letter $s$ in kinematics

  • Do not confuse this letter for speed!

  • Displacement is measured in metres

What is a displacement function?

• The displacement of an object, $s$ metres, can be written as a function of time, $t$ seconds

  • $s=\mathrm{f}\left(t\right)$

• Substitute a value of time in to find its displacement at that time

  • For example, $s=2t-t^2+1$

    • Initially, $t=0$ gives $s=2\times 0-0^2+1=1$ (1 metre in front of the origin)

    • After 3 seconds, $t=3$ gives $s=2\times 3-3^2+1=-2$ (2 metres behind the origin)

What is the velocity and how do I find it?

• Velocity is the speed and direction of an object

  • It is positive if moving forwards

  • It is negative if moving backwards

  • Do not confuse velocity and speed

    • Speed is always positive!

• To find the velocity of an object, $v$ metres per second, differentiate its displacement function

  • $v=\frac{\mathrm{d}s}{\mathrm{d}t}$

• For example, if $s=t^3-4t^2+2t$ then $v=3t^2-8t+2$ (by differentiation)

  • The initial velocity ($t=0$) is $v=3\times 0^2-8\times 0+2=2$ ms^-1

  • The velocity after 1 second ($t=1$) is $v=3\times 1^2-8\times 1+2=-3$ ms^-1

    • Its speed is 3 ms^-1

• If a velocity is zero at any point in time, it is said to be at instantaneous rest

  • It is stationary (not moving) at that instant in time

    • but not stationary all the time

  • To find the times at which the particle is at rest, set $v=0$ and solve to find $t$

What is the acceleration and how do I find it?

• Acceleration is rate at which the velocity changes

  • It is positive if speeding up (when moving forwards)

  • It is negative if slowing down (when moving forwards)

    • A negative acceleration is also called a deceleration

    • The magnitude of acceleration is always positive

• To find the acceleration of an object, $a$ metres per second per second, differentiate its velocity function

  • $a=\frac{\mathrm{d}v}{\mathrm{d}t}$

• For example, if $v=3t^2-8t+2$ then $a=6t-8$

  • You can substitute times in to find accelerations

• It the acceleration is always zero then the particle moves at a constant speed

How do I find the acceleration from the displacement?

• You differentiate displacement to get velocity, then differentiate velocity to get acceleration

  • So you differentiate displacement twice to get acceleration