3.5.2 Collisions

The principle of conservation of momentum states that for a closed system the total momentum in a specified direction remains constant as long as no external forces act on the system.

It is important to recognise that both momentum and total energy are conserved when objects collide. There are two types of collision:

• perfectly elastic:
• momentum is conserved, total kinetic energy is conserved.
• inelastic:
• momentum is conserved, total kinetic energy is not conserved.

As we know, linear momentum must always be conserved. We can resolve the different directions of momenta into 'x' 9horizontal) and 'y' (vertical) directions and ensure that each is constant.

3.5.1 Newton’s laws of motion

okay so Newton (v clever chap) has 3 laws of motion:

• First law: An object will remain at rest or continue to move with constant velocity unless acted upon by a resultant force.
• Second law: The net force acting on an object is directly proportional to the rate of change of momentum.
• Third law: When two objects interact they exert an equal and opposite force on each other.

The linear momentum of an object depends on its mass and velocity. We can calculate it using the equation:

p = m v

It has the unit kg m s^-1. It is a vector as it is a product of mass (scalar) and velocity (vector).

Newton's second law we can formulate the equation:

F = Δp/Δt

NOTE: The constant (k) in this equation can be made equal to 1 by defining the unit of force (the Newton) as the force required to give a 1kg mass an acceleration of 1m s^-2. It is also important to recognise that F = ma is a special case of this equation.

If we rearrange this equation we can get F Δt = Δp. The impulse of a force is defined as the product of force and the time for which this force acts on an object. It follows that the area under a force-time graph is equal to impulse.

2.2.1 Measurements and uncertainties

Okay so this is not exactly content in itself but is sort of like good information to know.

A true value is the value that would be obtained in an ideal measurement. A measurement error is the difference between a measured value and the true value for the quantity being measured. Mistakes are not counted as errors. If you make a mistake you can repeat an experiment without the mistake. However, an experiment will still contain errors no matter if you repeat it. In science, an error is the difference between the result you get and the correct result. They are usually caused by measuring devices even if they are used correctly. They can also be caused by the design of the experiment itself.

Random errors can happen when any measurement is being made. They are measurement errors in which measurements vary unpredictable. They cannot be corrected but we can reduce their effects by making more measurements and finding the mean value. Reasons for this include:

• factors that are not controlled in the experiment
• difficulty in deciding on the reading given by a measuring device.

Systematic errors are measurement errors in which the measurements differ from the true values by a consistent amount each time a measurement is made. Unlike random errors it is possible to correct systematic errors e.g by changing equipment. Reasons for this include:

• the way in which measurements are taken
• faulty measuring devices

Examples include poor contact between a thermometer and the object whose temperature is being measured, a faulty measuring device (e.g calibrated incorrectly/zero errors).

When we talk about obtaining measurements it is vital that we know the difference between precision and accuracy:

• precision regards how close repeated measurements are to each other (the closer together the more precise)
• accuracy regards how close a measurement result is to the true value (the closer the more accurate)

Uncertainties

Okay so because of random and systematic errors basically it is very hard to obtain the same value for a particular measurement. A mean value can be calculated by adding all the values and dividing it by the number of values. The range is the difference between the smallest and largest readings. The uncertainty in the measurement is an interval within which the true value can be expected to lie.

The absolute uncertainty in the mean value of a measurement can be approximated as half the range. It is expressed as a ± value. When you have a single measurement/repeat measurements are equal you approximate the absolute uncertainty to be equal to the resolution of the measuring instrument.

The percentage uncertainty can be calculated from its absolute uncertainty and mean values:

% uncertainty = 100 x absolute uncertainty/mean value

The final uncertainty depends on how quantities are combined:

• adding or subtracting quantities: add the absolute uncertainties for each value
• multiplying or dividing quantities: add the percentage uncertainties for each value
• raising to a power: increase the percentage uncertainty by the magnitude of the power

I'm sure you will have completed this in PAGs, but we can also use error bars/lines of best and worst fit. The absolute uncertainty in the gradient is the positive difference between the gradient of the line of best fit and the gradient of the line of worst fit. The percentage uncertainty can then be calculated using the following equation:

% uncertainty = 100 x absolute uncertainty/gradient of best fit line

3.3.3 Power

Power is the rate of work done (or the rate of energy transfer, since work done = energy transferred). We can determine power using the equation:

P = W / t

Power has the unit J s^-1 or W (watts). One watt is equal to one joule per second.

In lots of situations a constant force must exerted to maintain a constant speed. For example, the rate of work done (power) by the forward force provided by  cars engine is equal to the rate of work done against the frictional forces acting on the cat (so the net force on the car is 0). We can depict this as:

A constant force F moves the car a distance x in a time t

work done by the force W = F x

P = W/t = Fx/t

The speed of the car (v) is the rate of change of distance: v = x/t

P = Fv


Efficiency
Not all machines/processes convert all their energy into useful work. Lots of energy often is outputted as thermal energy (heat). We can calculate the efficiency of something using the following equation:

efficiency = 100% x useful output energy / total input energy

3.3.2 Kinetic and potential energies

Kinetic energy
Kinetic energy is the energy associated with the motion of an object. Kinetic energy (Ek) can be calculated from mass and speed:

Ek = 0.5 m v²

This means that, for a constant speed, Ek is directly proportional to mass. For a given mass, Ek is directly proportional to the square of the objects speed.

We need to  know how to derive this:

Take an object that initially starts at rest (u = 0). We can determine it's velocity at a certain distance using v² = u² + 2as:

s = (v²-u²)/2a = v²/2a

This means that the work done by the force moving the object is entirely transferred to Ek:

Work done = Ek = F x = F s

F = ma

Ek = ma s = ma (v²/2a)

Ek = mv²/2 = 0.5 m v²


Gravitational potential energy
Gravitational potential energy os the capacity for doing work as a result of an object's position in a gravitational field. You can calculate change in GPE from it's mass and height:

Ep = m g h

This one is a lot simpler to derive: When you lift something through a height h at a constant speed (no change in Ek) you have applied a force that equates to mg. The work done is transferred into Ep (GPE)

Ep = W = force x distance moved in the direction of the force

Ep = (mg) h

Ep = m g h


Also (when we cover gravitational fields):
The gravitational potential energy 'E' of any object with the mass m within a gravitational field is defined as the work done to move the mass from infinity to a point in a gravitational field:

E = m Vg

In a uniform gravitational field, in order to change the gravitational potential energy of an object its height above the surface must be changed as this results in a change in gravitational potential.

In a radial field (remember Vg = -GM/r) the gravitational potential energy can be written as:

E = m Vg = -GMm/r

Energy exchanges
Often, kinetic energy and gravitational energy will be exchanged. For example, if you drop a book it's gravitational energy will decrease whilst its terminal velocity increases:

0.5 m v² = m g h

0.5 v² = g h

v² = 2 g h

v = √(2 g h)

3.3.1 Work and conservation of energy

Work done is defined as the product of force and distance moved in the direction of the force. We can use the following equation to determine work done:

W = F x

(at an angle: W = F x cosθ)

It has the unit Nm. 1 Nm = 1J. One Joule is defined as the work done when a force of 1 N moves its point of application 1 m in the direction of the force.

It is important to understand that work done = energy transferred (this is because energy is defined as the capacity to do work). Energy is a scalar quantity with the unit Joule. The energy of all systems with mass can be classified as kinetic energy (energy due to the movement of an object) and gravitational potential energy (the energy due to the position of an object in the Earth's gravitational field). 

It is imperative to understand that the total final energy is always equal to the total initial energy. This can be represented in the principle of conservation of energy: the total energy of a closed system remains constant, energy can never be created nor destroyed but can be transferred from one form to another.

3.2.4 Density and pressure

Density
The density of a substance is defined as its mass per unit volume:

ρ = m / v

Unit: kg m^-13

Okay so it doesn't explicitly state in the spec that we need to know methods of determining density but I have a feeling we should so here goes. We need to measure the mass and volume of the object. Mass can be measured using a digital balance. We can calculate the volume of regular shapes (e.g a cuboid) by taking measurement with a ruler or we can submerge irregular shapes in a measuring cylinder of water and see how much the water is displaced. Now do ρ = m / v.

Pressure
Pressure is the normal contact force exerted per unit cross-sectional area:

p = F / A

Unit: N  m^-2

Archimedes' principle
We need to know about pressure in liquids and gases too, not just in solids. Gases and liquids exert pressure on surfaces because of the constant bombardment by their molecules.

We can calculate the pressure exerted by a vertical column of a liquid using:

p = hρg

The buoyant force on submerged objects can be explained in terms of pressure differences at its upper and lower surfaces.

Force at the top surface = hρgA
Force at the bottom surface = (h + x) ρgA
Resultant upward force = (h + x)ρgA - hρgA
Upthrust = xρgA

The volume of the object is Ax and it has mass Axρ so the upthrust is equal to the weight of the fluid displaced. This is Archimedes' principle, 'when an object is immersed fully or partially in a fluid, the upward force applied by the fluid on the object is the same as the weight of the fluid displaced by the object'.

If upthrust is less than weight the object will sink. For floating objects upthrust must equal the weight of the object.

3.2.3 Equilibrium

The moment of a force is the turning effect of a force about an axis or point. It equates to the product of force and the perpendicular distance of the line of action of the force from the axis or point of rotation:

moment = F x

Units: N m

When a body is in equilibrium the net force acting on it is zero and its net moment is zero. The principle of moments can be used to solve problems where an object is in rotational equilibrium. The principle of moments states that for a body in rotational equilibrium the sum of the anticlockwise moments abut any point is equal to the sum of the clockwise moments about the same point.

A pair of equal but opposite forces that are parallel and act on different lines is known as a couple. The moment of a couple is known as a torque. The torque of a couple is the product of one of the forces and the perpendicular separation between the forces:

torque of a couple = Fd

The centre of mass of an object is a point through which any externally applied force produces straight line motion with no rotation. The centre of gravity of an object is an imaginary point were the entire weight of an object appears to act. A freely suspended object will come to rest with its centre of gravity vertically below the point of suspension. We can use a plumb-line to experiment about the centre of gravity (for 2-D objects).

When an object is in equilibrium there is no resultant force acting on it. We can demonstrate the forces in a free-body diagram using a triangle of forces:

• arrows are drawn to represent each of the three forces end to end (ie the end of one arrow is the beginning of the next)
• the triangle is closed if the object is in equilibrium. this is because the et force is zero and so the object is in equilibrium.

3.2.2 Motion with non-uniform acceleration

If an object is moving through a fluid it will experience a drag force from the fluid. Drag is a frictional force that opposes the motion of the object. Its magnitude is mainly affected by:

• the speed of the object
• the cross-sectional area of the object
• the shape of the object
• the density of the fluid
• the texture of the object

Objects with larger cross-sectional areas will experience a greater drag force. This is because drag ∝ speed². 

Often, we call the drag force experienced by objects in air air resistance. We are able to reduce air resistance. For example, modern vehicles have streamlined shapes as this increases their top speed and reduces their fuel consumption.

During a vertical fall (e.g through air) the weight of an object will of course remain constant but the speed will increase so the drag force will increase. At the instant objects start to fall the object will experience no drag force and the total force is equal to the weight (as this is the only force acting on the object). As the object falls its speed increases so drag increases. The resultant force on the object decreases (as drag opposes the motion ie is in the opposite direction to weight). At terminal velocity, drag = weight and the object has zero acceleration and its speed is constant.

We can investigate the motion of an object falling affected by drag force by using a motion sensor and data-logger:

• attach the object to a light polystyrene ball using a thin thread passing over a pulley.
• Drop the object through a cylinder of liquid (e.g water or glycerol). This will pull the polystyrene ball vertically upwards (the motion of this ball is equal to the motion of the object)
• when the ball is moving up with constant velocity, the object is moving down with terminal velocity

3.2.1 Dynamics

The mass of an object depends on the amount of matter it contains. The net/resultant force acting on an object will make it accelerate in that particular direction. The net force (F):

F = m a

Force is measured in newtons (N). A force of 1 newton will give a 1kg mass an acceleration of 1m s^-1 in the direction of the force.

The weight of an object on the surface of the Earth is the gravitational force acting on the object:

W = m g

The weight of an object can be calculated using a newtonmeter. These are calibrated to show the gravitational force acting on an object in newtons (e.g an object of mass 1kg will show a weight of 9.8N).


The easiest way to analyse the forces acting on an object is to draw free-body diagrams (these isolate all the forces acting on a particular object. Each force vector is represented by an arrow labelled with the force it represents, each arrow is drawn to the same scale (ie the longer the arrow the greater the force).

Tension, normal contact force, upthrust, and friction are all examples of forces that can be added to a free-body diagram.

3.1.2 Linear motion

Okay so we need to learn 4/5 of the suvat equations (no. 1, 2, 3, and 4) :

Tbh, number 5 is useful if you want to learn it, but you can use the others to deduce the answer so not the end of the world (you do need to know no.5 if you're doing maths though).


Okay so one experiment we need to know about is about free fall and determining g (the gravitational field strength), acceleration due to gravity. As we know, objects with mass exert a gravitational force on each other. When an object is accelerating under gravity (with  no other force acting on it) it is said to be in free fall. As I mentioned just a few moments ago, we need to be able to determine g (in a laboratory):

• Hold a small steel ball over a trapdoor using an electromagnet
• Turn the current off, a timer will trigger and the electromagnet will demagnetise (hence the ball will fall)
• the heavy steel ball will drop over a known distance (measure with a meter rule)
• it's decent will be timed using the timer (use a stopwatch for backup)
• Calculate the value for g using a = v/t = g

We can also use light gates:

• Two light beams are placed one above the other with detectors connected to a timer
• When the ball falls through the first beam it interrupts the light and starts the timer
• When the ball falls through the second beam a known distance further down the timer will stop
• use a = v/t = g

We can also use pictures:

• Drop a small metal ball from rest next to a metre rule
• Record its fall with a camera in rapid0-fire repeating mode
• Hold the camera shutter open to produce a photograph with multiple images of the falling ball/put a long exposure setting on
• Examine the photograph and determine the position of the ball at regular time intervals
• do a = v/t

Okay so we also need to know a little bit about things such as reaction time, thinking distance breaking distance, and stopping distance for a vehicle. Basically:

• The stopping distance is the total distance travelled from when the driver first sees the reason to stop to when the vehicle stops. It is thinking distance + breaking distance
• Thinking distance is the distance travelled between the moment when you first see a reason to stop to the moment you use the brake. For a vehicle moving at constant speed thinking distance = speed x reaction time. The greater the speed or reaction time the further the vehicle will travel before its driver applies the brakes. 
• Breaking distance is the distance travelled between the time the brake is applied until the vehicle stops.

NOTE: One point in this section of the spec states that we need to know '(ii) techniques and procedures used to investigate the motion and collisions of objects'. You should have done this as a PAG experiment (usually in your first year of covering the course).

3.1.3 Projectile motion

How far projectiles can travel depends on several factors. It is important to realise that the vertical and horizontal motions of the ball are independent of each other. Assuming no air resistance:

• the vertical velocity changes due to acceleration of free fall
• the vertical displacement and time of flight can be calculated using the equations of motion
• horizontal velocity remains constant

The horizontal velocity of the projectile is constant because acceleration and velocity are vectors. The acceleration of free fall is vertically downwards and the component of this acceleration in the horizontal direction is zero. This is because horizontal acceleration = g cos 90° = 0.

The path of the projectile is curved because the vertical component of its velocity increases with time whilst the horizontal component is unaffected. The magnitude of the actual velocity (v) of the projectile can be calculated from he vertical and horizontal components of the velocity (vx and vy):

v = √((vx)² + (vy)²)

The angle made by the velocity to the horizontal is given by:

θ = tan^-1 ((vy)/(vx))

When a projectile is fired upwards the motion can still be analysed in terms of the independent motion in the horizontal and vertical directions. The horizontal component of the velocity if vcosθ and the initial vertical component of the velocity if vsinθ.

5.4.4 Gravitational potential and energy

Gravitational potential
The gravitational potential (Vg) at a point in a gravitational field is defined as the work done per unit mass to move an object to that point from infinity. It has the units J kg^-1 and infinity refers to a distance so far from the object producing the gravitational field that the gravitational field strength is zero. Gravitational potential is a scalar quantity. At infinity gravitational potential is at a maximum 0J kg^-1. Since all masses attract each other it takes energy (external work must be done) to move objects apart.

The gravitational potential at any point in a radial field around a point mass depends on:

• the distance 'r' from the point mass producing the gravitational field to that point
• the mass M of the point mass.

We can use the ideas of work done on a force and Newton's law of gravitation to form the equation:

Vg = -GM/r

A graph of Vg against 1/r will produce a straight line through the origin with a gradient equal to -GM. If a graph is drawn of the Vg around the Earth against distance r from the centre of mass of the Earth, the smallest value of r on the graph will be the Earths radius.

Gravitational potential energy
The gravitational potential energy 'E' of any object with the mass m within a gravitational field is defined as the work done to move the mass from infinity to a point in a gravitational field:

E = m Vg

In a uniform gravitational field, in order to change the gravitational potential energy of an object its height above the surface must be changed as this results in a change in gravitational potential.

In a radial field (remember Vg = -GM/r) the gravitational potential energy can be written as:

E = m Vg = -GMm/r

Force-distance graphs
We know, from Newton's law of gravitation, that the gravitational force between objects decreases with distance in an inverse-square relationship. The area under a force-distance graph is equal to the work done (remember this can be negative if the object falls as the object loses gravitational potential energy).

Escape velocity
In order to escape the gravitational field of a mass an object must be supplied with energy equal to the gain in gravitational potential energy needed to lift it out of the field:

1.5mv² = GMm/r

The minimum velocity for this condition to be met is known as the escape velocity. It is the same for all objects regardless of their mass:

v = √(2GM/r)

5.4.3 Planetary motion

Kepler had three different laws to help us with planetary motion:

• K1 - the orbit of a planet is an ellipse with the Sun at one of two foci
• In most cases orbits have low eccentricity so their orbits are modelled as circles. E.g the difference is distance from aphelion (furthest point from the Sun) and perihelion (closest point from the sun) is just 3%
• K2 - A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time
• As plants move on their elliptical orbit around the Sun their speed is not constant. When a planet is closer to the Sun it moves faster. This explains why we don't see great comets - their orbits are highly elliptical and when they get close to the sun they move fast and spend much less time on this part of their orbit. They also spend most of their time too far from the Sun to be visible.
• K3 - The square of the orbital period T of a planet is directly proportional to the cube of its average distance r from the sun.
• T² ∝ r³ ∴ T²/r³ = K

Most planets in the Solar System have almost circular orbits so we can use circular motion mathematics along with Newton's law of gravitation to relate the orbital period 'T' of a planet to it's distance 'r' from the Sun, justifying K3...

If we think about it, a planet of mass m orbiting the Sun at a distance r will have a speed of v and an orbital period of T. The mass of the sun is M (M_☉) and the centripetal force on the planet is provided by the gravitational force between it and the Sun. Therefore the gravitational force 'F' on the planet is equal to the centripetal force:

F = mv²/r = GMm/r²

v² = GM/r

The planet is moving in a circle. This means the speed of the planet can be determined by dividing the circumference of its orbit by its orbital period (v = 2πr/T). We can substitute this into the above equation:

4π²r²/T² = GM/r

T² = (4π²/GM)r³

This equation can be used to determine the mass of an object, the orbital period of a planet, or its distance from the Sun. From this we can make the following statements:

• The ratio T²/r³ is equal to 4π²/GM
• The gradient of the graph of T² against r³ is equal to 4π²/GM

It is important to note that Kepler's laws also apply to any smaller objects in orbit around a larger one (e.g satellites and moons in orbits around planets) and not just planets in our Solar System.

Geostationary satellites

Satellites orbiting the Earth obey Kepler's laws of planetary motion. Satellites always fall to Earth due to the fact that the only force acting on them is the gravitational attraction between it and the Earth. However it is travelling at such a great distance that as it falls the Earth curves away beneath it so it stays at the same height above the surface. All satellites must be at a given height and speed to keep a stable orbit. We can rearrange the above equation to form an equation for the speed 'v' of a satellite in stable orbit at a distance 'r' from the centre of the Earth:

v = √(GM/r)

NOTE: the mass 'm' of the satellite is not present meaning all satellites placed in a given orbit at a given height will travel at the same speed.Once launched they are usually above the atmosphere so they experience no air resistance to slow them down so their speed remains constant.

There are a few uses of satellites which are pretty useful (duh):

• communications: satellite phones (NOT mobile phones!!), TV, some types of satellite radio
• military uses: reconnaissance
• scientific research: looking down on Earth to monitor crops/pollution/vegetation etc, looking outwards to study the Universe
• weather and climate: predicting/monitoring the weather across the globe. Also monitoring long-term changes in climate
• global positioning

Depending on the use, different satellites have different orbits. Satellites placed in the equator can be geostationary satellites. As the height of these satellites increase, so does their period. A geostationary satellite is placed in a geostationary orbit so it remains above the same point of the Earth whilst the Earth rotates. To be a geostationary satellite the satellite must:

• be in an orbit above the Earth's equator
• rotate in the same direction as the Earth's rotation
• have an orbital period of 24 hours

5.4.2 Newton’s law of gravitation

Newton's law of gravitation can also be described as a universal law of gravity. It describes the forces between any objects that have mass. The law can be used to explain motions such as the motion of planets round a sun, and why objects near the surface of the Earth will fall to the ground.

If we think about it, two objects m and M separated by a distance r will each create it's own gravitational field. The interaction of these fields gives rise to forces between the objects. According to N3 the objects must experience an equal and opposite force. Newton's law of gravitation states the force between two point masses is:

• directly proportional to the product of the masses, F ∝ Mm
• inversely proportional to the square of their separation, F ∝ 1/r².

From these statements we can create the equation:

F ∝ Mm/r²

The constant in this equation is known as the gravitational constant 'G'. This value is 6.67408 × 10^-11 m³ kg^-1 s^-2:

F = -GMm/r²

NOTE: we must use a minus sign as this shows that gravitational force is an attractive force. The attractive force (F) decreases with distance in an inverse-square relationship (F ∝ 1/r²) so doubling the distance will decrease the force by a factor of 4 (2²).

If several objects are involved the resultant force can be determined by vector addition.

5.4.1 Point and spherical masses

Firstly, it is important to understand that ALL objects with mass create a gravitational field. This field extends to infinity but gets weaker with distance from the centre of mass of the object. Any object placed in a gravitational field will experience an attractive force towards the centre of mass of the object that is creating the field (e.g for objects on Earth we call this gravitational attraction the objects weight).

Gravitational field strength
The gravitational field strength 'g' at a point within a gravitational field is defined as the gravitational force exerted per unit mass on a small object placed at that point within the field. It is a vector and always points to the centre of mass of the object creating the gravitational field. It can be calculated using the equation:

g = F/m

NOTE: F is the gravitational force and m is the mass of the object in the gravitational field. Gravitational field strength has the unit  N kg^-1. This unit is the same as m s^-1. This is because gravitational field strength at a point is the same as the acceleration of free fall of an object at that point (g = a).

Gravitational field patterns
We can map the gravitational field pattern around an object by using gravitational field line/lines of force. The lines never cross and always show the direction of the field (the direction of force on a mass at that point in a field). Field lines that are closer together indicate a stronger field. The field lines around a spherical mass form a radial field. The gravitational field strength decreases with distance from the centre of the mass (the field lines get further apart). We can model large planets as point masses with field lines converging at the centre of mass of the object because the radial fields for a spherical mass and a single point mass are very similar. If the field lines are parallel and equidistant the field is uniform. In a uniform field the gravitational field strength does not change.

Gravimetry
This isn't exactly in the spec but we've covered it so I thought it might be useful to cover on the blog:

Gravimetry is the precise measurement and study of a gravitational field. Earth's gravitational field can be mapped and minute variations detected (usually due to topography, e.g. mountains, craters from meteorites etc). Gravimetry is used in mineral prospecting as denser rock causes a higher than normal gravitational field on the Earths surface (indicating metal ores are present there). We also use this technique to search for oil as oil-bearing rocks have a lower density than the surrounding rock.

Superconducting gravimeters are the most accurate gravimeters. They use liquid helium to cool a superconducting sphere in a magnetic field. The weight of the field is balanced by the effects of the magnetic field. The electric current required to generate the magnetic field depends on the Earth's gravitational field strength at that point.

Different fields
It is important to remember that not just gravitational fields give rise to a force. We also have magnetic, and electric fields we cover in this spec!

5.5.3 Cosmology

So in this section we'll use different unit for distance (as apposed to the meter). This is because the distances are so vast. The new units are as follows:

• Astronomical unit (AU)
• This is the average distance from the Earth to the Sun. It is most often used to express the average distance between the Sun and other planets in our Solar System.
• It is equal to 1.50 x 10¹¹ m
• Light-year (ly)
• This is the distance travelled by light in a vacuum in a time of one year. It is often used when expressing distances to stars or other galaxies.
• It is equal to 3.00 x 10⁸ x (365 x 24 x 60 x 60) = 9.46 x 10¹⁵ m
• Parsec (pc)
• So for this one, we need to be aware that in cosmology we measure angles in arcseconds and arcminutes. One arcsecond is 1/3600th of a degree (1 arcminute is 1/60th of a degree).
• The parsec is defined as the distance at which a radius of one AU subtends an angle of one arcsecond.
• The value of 1pc can be determined by tan(1/3600) = 1AU/1pc. This means that 1pc = 1AU/tan(1/3600). This equals 3.1 x 10¹⁶ m.
• It is important to realise that at a distance of n parsecs, the angle subtended by a radius of 1AU = 1/n arcseconds.

There is a special technique used to determine the distance to stars that are relatively close to Earth (like, less than 100 pc). This is known as stellar parallax. Parallax is the shift in position of a relatively close star against the backdrop of much more distant stars as Earth orbits the Sun. If p (parallax angle) is measured in arcseconds, the distance to the nearby star in parsecs is given by the following equation:

d = 1/p

This technique can't be used to measure the distance between stars bigger than 100pc from the Earth because as d increases the parallax angle decreases. Eventually becoming too small to measure accurately, even with the most advanced astronomical techniques.

The Doppler effect
The Doppler effect is used to determine the speed of moving objects.

When a wave source moves relative to an observer, the frequency and wavelength of the waves received by the observer change compared with what would be observed without relative motion. Originally, two points equidistant from the source would receive waves at the same frequency and wavelength as they were emitted from the source. When the source moves closer to one point, the waves received by this point will be compressed. They have a shorter wavelength and a higher frequency (therefore, a shorter period).

How fast the wave source moves relative to the observer affects the size of the observed shift in wavelength and frequency. For electromagnetic waves we can use the Doppler equation. The equation shows that the faster a source moves, the greater the observed change in wavelength and frequency:

Δλ/λ ≈ Δf/f ≈ v/c

NOTE: The Doppler equation can only be used for galaxies with speed far less than the speed of light.

As we know already, one technique to analyse starlight involves looking at the absorption spectra from stars. The Doppler effect can be used to determine the relative velocity of a distant galaxy. Any difference in the observed wavelengths of the absorption lines must be caused by the relative motion between the galaxy and the Earth.

NOTE: if the galaxy is moving towards the Earth the absorption lines will be blue-shifted (they move toward the blue end of the spectrum because the wavelength appears shorter). If the galaxy is moving away from the Earth the absorption lines will be red-shifted as the wavelength appears stretched.

Using data from the absorption spectra of many distant galaxies, Hubble made two key observations:

1. A confirmation that earlier observations that the light from the vast majority of galaxies was red shifted (they had a relative velocities away from the Earth)
2. He found that in general the further away the galaxy was the greater the observed red shift and so the faster the galaxy was moving

From these observations Hubble formulated his law (Hubble's Law): The recessional speed (v) of a galaxy is almost directly proportional to its distance (d) from the Earth.

This means that a graph of recessional speed against distance for all galaxies will produce a straight line graph through the origin. The gradient is a constant of proportionality - the Hubble constant (Ho). It's (current) value is 2.2  × 10^-18 km s^-1 Mpc^-1. From Hubble's law it can be derived that:

v ≈ Ho x d

Hubble's law has been very useful in determining key evidence for the Big Bang theory and the model of the expanding Universe (following the Big Bang). This model is the accepted explanation of the observation that the light from nearly all the galaxies we can see is red-shifted. The further two points are apart the faster their relative motion (the more red-shifted their spectra).

The cosmological principle is the assumption that (when viewed on a large enough scale) the Universe is homogenous and isotropic and the laws of physics are universal:

• Homogenous means the matter is distributed uniformly across the Universe (ie the density of the universe is uniform
• Isotropic means that the Universe looks the same in all directions to every observer (ie there is no end to the Universe)
• The laws of physics can be applied across the Universe (meaning that theories/models tested on Earth can be applied to everything within the Universe).

The Big Bang
Hubble's law and the microwave background radiation are two key pieces of evidence for the Big Bang theory. Hubble's law shows that space is expanding as the galaxies are receding from each other because space itself is expanding in all dimensions.

The existence of microwave background radiation is the second piece of evidence for the Big Bang. Microwave background radiation can only be explained by the Big Bang and the expansion of space. It's existence can be explained in two ways:

• When the Universe was young and extremely hot space was saturated with high-energy gamma photons. The expansion meant that space itself was stretched over time. The expansion stretched the wavelength of these high-energy photons so we now observe this primordial electromagnetic radiation as microwaves.
• The Universe was extremely hot and dense when it was young. Expansions over billions of years reduced it's temperature to about 2.7 K. The Universe can be treated as a black-body radiator. At this temperature the peak wavelength would correspond to about 1mm (in the microwave region of the spectrum).

We can estimate the age of the Universe by assuming that it has expanded uniformly over time since the Big Bang. This actually isn't the case lol. Results from recent observations show that the expansion of the Universe is accelerating. Nonetheless, this assumption will give a crude indication the Universe's age.

So, Hubble's law shows galaxies are receding from each other. If a galaxy at a distance 'd' is moving away at a constant speed 'v' then a time (d/v) must have elapsed since it was next to our galaxy. This time is roughly the age of the Universe. The ratio d/v is equal to 1/Ho meaning that:

age of the Universe 't' ≈ 1/Ho

NOTE: This gives the age of the Universe to be 4.5 x 10¹⁷ s (≈14 billion years).

As I mentioned above, it is now known that the Universe appears to be expanding at an increasing rate. The most widely accepted theory includes the concept of dark energy. It is suggested that this hypothetical form of energy fills all of space and tends to accelerate the expansion of the Universe. The two most significant discoveries that have changed our understanding of the Universe is the discovery of dark energy and dark matter.

We need energy to accelerate things. The term 'dark energy' was coined to describe a hypothetical form of energy that permeates all space. It is estimated that dark energy makes up around 68% of our Universe.

In the 1970s astronomers studying the Doppler shift in light from galaxies found that the velocity of the stars in the galaxies did not behave as predicted. It was expected that their velocity would decrease as the distance from the centre of the galaxy increases. This effect is observed in other gravitational systems where most of the mass is in the centre (e.g the moons of Jupiter). The observations can be explained if the mass of the galaxy is not concentrated in the centre. We currently think that there must be another type of matter which we cannot see. This dark matter is spread throughout the galaxy, explaining the observations. The Universe must be made up of 27% of this matter (according to calculations).

All we know about dark matter is we know it cannot be seen directly with telescopes and it neither emits not absorbs light.

The rest of the universe is made up of a small percentage of ordinary matter.