Kinematic concepts

2.1.1 Define displacement, velocity, speed and acceleration.

• Vector quantities always have a direction associated with them.

2.1.2 Define and explain the difference between instantaneous and average values of speed, velocity and acceleration.

• Average value - over a period of time.

• Instantaneous value - at one particular time.

2.1.3 '"Outline the conditions under which the equations for uniformly accelerated motion may be applied.
Acceleration must be constant

Graphical representation of motion

2.1.4 Draw and analyse distance–time graphs, displacement–time graphs, velocity–time graphs and acceleration–time graphs.

2.1.5 Analyse and calculate the slopes of displacement–time graphs and velocity – time graphs, and the areas under velocity–time graphs and acceleration–time graphs. Relate these to the relevant kinematic quantity.

Uniformly accelerated motion

Determine the velocity and acceleration from simple timing situations

[a = (v - u)÷t] [s = (u + v)t÷2] [s = ut + 0.5at^2] [v^2 = u^2 + 2as]

Derive the equations for uniformly accelerated motion.There are 3 equations for uniformly accelerated motion. The first one is derived form the definition of acceleration as the rate of change of velocity,

i.e

             acceleration=velocity change/time taken. 

Let the initial and final velocities be designated by u and v respectively, acceleration by a and time by t, then

                          a=(v-u)/t.

Making v the subject we get,

                         v=u+at,

which is the first equation of motion

Describe the vertical motion of an object in a uniform gravitational field.

Describe the effects of air resistance on falling objects.

Solve problems involving uniformly accelerated motion.

Forces and free-body diagrams

Identify the forces acting on an object and draw free-body diagrams representing the forces acting. Each force should be labelled by name or given a commonly accepted symbol. Vectors should have lengths approximately proportional to their magnitudes.

Newton’s first law

Newton's First Law of Motion states that a body will remain at rest or moving with a constant velocity unless acted upon by an unbalanced force.

Equilibrium

Equilibrium is the condition of a system in which competing influences (such as forces) are balanced.

Newton’s second law

ΣF = ma

Alternately: ΣF = Δp/Δt In words, the resultant force is all that matters in the second law. The direction of motion depends on the direction of the resultant force.

Newton’s third law

If body A exerts a force on body B, then body B exerts an equal and opposite force on body A.

Essentially every action force has an equal and opposite reaction force. This is why you are able to place your coffee mug on the table without it falling into the table. The table provides a normal reaction force (up) to counter the gravitational force (down) hence F(net) = 0 and your coffee mug stays on the table.

Work

Work refers to an activity involving a force and movement along the direction of the force. It is a scalar quantity that is measured in Joules (Newton meters in SI units) which can be defined as:

Work done= F×s×cosθ

Where F is the force applied to the object, s is the displacement of the object and cosθ is the cosine of the angle between the force and the displacement. In a linear example (with the force being exerted in the same direction as the displacement), the cosθ is equal to 1 and the equation simplifies to ${\displaystyle Work=F\times s}$.

Example calculation: If a force of 20 newtons pushes an object 5 meters in the same direction as the force what is the work done?

F= 20 N
s=5 m
W=F×s=20×5= 100 J 
100 Joules of work is done

Examples (when is work done?):

Force making an object move faster (accelerating)
Lifting an object up (moving it to a higher position in the gravitational field)
Compressing a spring

When is work not done

When there is no force
Object moving at a constant speed
Object not moving

Some useful equations;

If an object is being lifted vertically the work done to it can be calculated using the equation

Work done= mgh

Where m is the mass in kilograms, g is the earth's gravitational field strength (10N kg^-1), and h is the height in meters

Work done in compressing or extending a spring

Work done = ½ kx²

Where k is Hooke's constant and x is the displacement

Energy and Power

Energy is the capacity for doing work. The amount of energy you transfer is equal to the work done. Energy is a measure of the amount of work done, this means that the units for energy and work must be the same- joules. Energy is like the "currency" for performing work. To do 100 joules of work, you must expend 100 joules of energy.

Conservation of energy

In any situation the change in energy must be accounted for. If it is 'lost' by one object it must be gained by another. This is the principle of conservation of energy which can be stated in several ways:

The total overall energy of a closed system must be constant

Energy is neither created or destroyed, it just changes form.

there is no change in the total energy of the universe

Energy can be in many different types these include:

Kinetic energy, Gravitational potential energy, Elastic potential energy, Electrostatic potential energy, Thermal energy, Electrical energy, Chemical energy, Nuclear energy, Internal energy, Radiant energy, Solar energy, and light energy.

You will need equations for the first three

Kinetic energy = ½ mv² where m is the mass in kg, v is the velocity (in ms^-1)

Gravitational potential energy= mgh where m is the mass in kg, g is the gravitational field strength, and h is the change in height

Elastic potential energy =½ kx² where k is the spring constant and x is the extension

Power -measured in Watts (W) or Joules per second (Js-1)- is the rate of doing work or the rate at which energy is transferred.

Power= energy transferred÷time taken= energy transferred÷time taken

If something is moving at a constant velocity v against a constant frictional force f, the power P needed is P= fv

If you do 100 joules of work in one second (using 100 joules of energy), the power is 100 watts.

Efficiency is the ratio of useful energy to the total energy transferred .

Work-Energy Principle

The change in the kinetic energy of an object is equal to the net work done on the object.

${\displaystyle W_{net}={\dfrac {1}{2}}mv_{final}^{2}-{\dfrac {1}{2}}mv_{initial}^{2}}$

This fact is referred to as the Work-Energy Principle and is often a very useful tool in mechanics problem solving. It is derivable from conservation of energy and the application of the relationships for work and energy, so it is not independent of the conservation laws. It is in fact a specific application of conservation of energy. However, there are so many mechanical problems which are solved efficiently by applying this principle that it merits separate attention as a working principle.

For a straight-line collision, the net work done is equal to the average force of impact times the distance traveled during the impact.

Average impact force x distance traveled = change in kinetic energy

If a moving object is stopped by a collision, extending the stopping distance will reduce the average impact force.