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   Stationary in front of the origin   Stationary behind the origin   Constant Velocity In a forwards (+) direction   Constant Velocity In a backwards (-) direction   Constant Acceleration   Constant Deceleration   Non-Uniform Acceleration   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.