PHY C3: Kinematics

Today we’ll be covering:

  • distance, displacement, speed, velocity, acceleration
  • graphs of motion
    • determine displacement from the area under a velocity-time graph
    • determine velocity using the gradient of a displacement-time graph
    • determine acceleration using the gradient of a velocity-time graph
  • equations of motion
  • uniformly accelerated motion in a straight line

Let’s get to it!


Some Basic Definitions

  • Distance: total path length covered (where direction does not matter)
  • Displacement: linear path length measured from initial position to finishing position
  • Speed: rate of change of distance
  • Velocity: rate of change of displacement
  • Acceleration: rate of change of velocity

When dealing with changing velocities & accelerations, it is important to distinguish between:

AVERAGE velocity/
acceleration
change in displacement/velocity over total time
INSTANTANEOUS velocity/
acceleration
change in displacement/velocity over a SMALL interval in time
(expressed on a graph as the gradient of a TANGENT)

 


Graphs of Motion
The motion of an object can be expressed as graphs of different values against time.

Behaviour of object

Displacement-Time Velocity-Time Acceleration-Time

Stationary at Origin

Picture1.png Picture2.png Picture3.png

Stationary in front of the origin

Picture4.png Picture2.png Picture3.png

Stationary behind the origin

Picture5.png Picture2.png Picture3.png

Constant Velocity
In a forwards (+) direction

Picture6.png Picture7.png Picture3.png

Constant Velocity
In a backwards (-) direction

Picture8.png Picture9.png Picture3.png
Constant Acceleration Picture10.png Picture11.png Picture12.png

Constant Deceleration

Picture13.png Picture14.png Picture15.png

Non-Uniform Acceleration

Picture16.png Picture17.png Picture18.png

Finding the gradient & area below these graphs can also show the relationships between the aspects of motion:

Calculation

Displacement-Time Velocity-Time

Acceleration-Time

Gradient

Velocity Acceleration

Area beneath graph

Displacement

Velocity


Uniformly Accelerated Motion
After understanding these basics, it’s useful to know the derivations of a group of equations that can be a handy shortcut when dealing with kinematics problems.

However, remember that they work only for UNIFORMLY accelerated motion.
The following equations are only applicable in a situation where..

  • the acceleration of the object is CONSTANT
  • the object is travelling LINEARLY (in 1 dimension)

There are other types of motion wherein these equations do NOT apply:

  • Circular Motion
  • Simple Harmonic Motion (Oscillations)

Since in these cases the object moves in 2 dimensions.


Kinematic Equations
Due to them containing the variables s, u, v, a & t,
these equations are commonly called the “SUVAT Formulas” by laypeople.

The equations are:

v = u + at

s = ½ (u + v)t

s = ut + ½ at2

s = vt – ½ at2

v2 = u2 + 2as

 


Derivations:

Let’s start with a basic equation for acceleration, which is change in velocity over time.
a = (v – u)/tThis can be rearranged to
v = u + at
Next, think about the definition of AVERAGE VELOCITY:

Total distance over total time
vavg = s/t

It is also known that average velocity is the sum of the initial & final velocity, divided by 2:
vavg = (u + v)/2

Thus,
s/t = (u + v)/2
s = ½ (u + v)t

 

Next, the 1st equation can be substituted into this one to get

s = ½ (u + u + at)t

s = ut + ½ at2

 

From v = u + at, you can rearrange to get

u = v – at

s = (v – at)t + ½ at2

s = vt – at2 + ½ at2

s = vt – ½ at2

 

From v = u + at, you can rearrange to get
t = (v – u)/aSubstituting this into s = ut + ½ at2s = u(v – u)/a + ½ a[(v – u)/a]2s = (uv – u2)/a + (v – u)2/2a2as = 2uv – 2u2 + (v – u)22as = 2uv – 2u2 + v2 – 2uv + u2

2as = u2 + v2

v2 = u2 + 2as

 

Before solving a question, ALWAYS:

  • Identify what you HAVE
  • Identify what you NEED
  • Identify the appropriate EQUATION to use

 


Free Fall Acceleration
On Earth, all objects are subject to a Gravitational Force.
This force accelerates all objects by 9.81 m s-2 downwards (towards the Earth).

This value of acceleration is known as the constant, g.

This acceleration must be taken into account when dealing with the motion of falling objects.

In Free Fall, g is the ONLY VERTICAL force acting upon the object.
This means that the only source of acceleration is the Earth’s gravity, not any thrusters or jets on the object.

 


Air Resistance
For many problems, air resistance is NEGLIGIBLE.
You won’t have to worry about taking this into account.

However, once you cover Air Resistance in the next chapter,
there will be problems where you have to take it into account.

You can read up on Air Resistance here.


⇐ Previous in Physics: Measurements & Uncertainties
⇒ Next in Physics: Projectile Motion

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