Download Space by Jonathan Chan

Space 1
Space
Part 1: The Earth’s Gravitational Field
Effects of the Earth’s Gravitational Field
Weight is a force on an object due to a gravitational field (𝑊 = 𝑚𝑎𝑔 ). Gravitational potential
energy is mass by acceleration due to gravity by the height above the surface (𝐸𝑃 = 𝑚𝑔ℎ), while
work done is the force by the displacement in the direction of the force (𝑊 = 𝐹𝑟). As F = ma and
height is the displacement if raising an object in a gravitational field, the formula becomes 𝑊 =
𝑚𝑎𝑔 ℎ. Thus, gravitational potential energy is related to work done.
Gravitational potential energy is the work done to move an object from a very large distance away
to a point in a gravitational field. The gravitational potential energy is greater when the object is
further from the Earth. Infinity is the maximum of gravitational potential energy, but the attraction is
so small that it is defined as zero. Hence, all the values of gravitational potential energy closer to the
𝐺𝑚 𝑚
Earth are all negative (𝐸𝑝 = − 𝑟1 2)
Space 2
Part 2: A Successful Rocket Launch
Projectile Motion
The trajectory of a projectile is the path it takes during its flight, which without air resistance should
trace out a parabola, given that gravity is constant. Projectile motion can be resolved into an
independent vertical motion and horizontal motion superimposed on each other.
Galileo proved that falling objects accelerate at the same rate, meaning that they have the same
acceleration due to gravity, but can be affected by air resistance. He suggested that vertical motions
are perpendicular to and independent of horizontal motions that are constant.
Escape Velocity
Escape velocity is the initial velocity required by a projectile to rise vertically and just escape the
planet’s gravitational field. As the projectile rises, the initial kinetic energy it has, is converted into
the final gravitational potential energy.
2𝐺𝑀𝑝𝑙𝑎𝑛𝑒𝑡
𝐸𝑠𝑐𝑎𝑝𝑒 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 = √
𝑟𝑝𝑙𝑎𝑛𝑒𝑡
, therefore it depends on the mass and radius of the planet.
Newton suggested a hypothetical situation where a projectile is
launched fast enough to achieve an orbit around the Earth. His
reasoning was that when thrown, a stone follows a parabolic path
and covers a considerable range before it hits the ground, so
theoretically, if thrown fast enough, as the stone falls, the Earth’s
surface curves away, so the stone orbits, rather than landing. If
launched even faster, the object could escape the Earth entirely.
G-Forces
The term g-force is used to express a person’s apparent weight as a multiple of their actual weight.
This is used to explain the forces acting on an astronaut during launch as they experience large
forces that significantly increase their apparent weight.
Launching a Rocket
Rockets are usually launched towards the east, in the direction of the Earth’s rotation to receive a
1700kmh-1 (460ms-1) boost in their velocity relative to the Sun to reduce the amount of fuel required
to achieve the target velocity. Engineers can exploit an extra 107000kmh-1 (30kms-1) from the Earth’s
orbit around the Sun for missions further into space by launching when the Earth’s orbital velocity
corresponds to the desired direction of launch.
Space 3
As the mass of the rocket decreases, its velocity and acceleration both increase. When blasting off
𝐹𝑛𝑒𝑡 = 𝑡ℎ𝑟𝑢𝑠𝑡 − 𝑚𝑔, so if m is decreasing and F is constant, a must be increasing. Rocket propulsion
is derived from the backward force of the rocket on the exhaust gases and the forward force of the
gases on the rocket. Since momentum is conserved:
𝑝𝑖 = 𝑝𝑓 = 0
0 = 𝑝𝑒𝑥ℎ𝑎𝑢𝑠𝑡 + 𝑝𝑟𝑜𝑐𝑘𝑒𝑡
∆𝑝𝑒𝑥ℎ𝑎𝑢𝑠𝑡 = −∆𝑝𝑟𝑜𝑐𝑘𝑒𝑡
∆(𝑚𝑣)𝑒𝑥ℎ𝑎𝑢𝑠𝑡 = −∆(𝑚𝑣)𝑟𝑜𝑐𝑘𝑒𝑡
−∆(𝑚𝑣)𝑒𝑥ℎ𝑎𝑢𝑠𝑡
∴ 𝑣𝑟𝑜𝑐𝑘𝑒𝑡 =
∆𝑚𝑟𝑜𝑐𝑘𝑒𝑡
As the acceleration of the rocket increases, the forces the astronauts experience increase.
Astronauts on the space shuttle endure 3g, while those on the Apollo missions experienced 6g.
Acceleration
2g
3g
4-6g
9-12g
Body Orientation
Upright and parallel to
the acceleration
Upright and parallel to
the acceleration
Upright and parallel to
the acceleration
Reclining and
perpendicular to the
acceleration
Effect
Walking becomes arduous
Walking is impossible
Gradual dimming of vision due to a decrease in blood
flow to retina. Eventually blacks out
Chest pain, fatigue and some loss of side vision, still
conscious and can move hands and fingers
Circular motion
Uniform circular motion is circular motion with a uniform orbital speed. Any object in circular
motion has a centripetal force that keeps it there and is directed to the centre of the circle.
Circular motion
Source of centripetal force
Whirling rock on a string
The tension in the string
Electron orbiting atomic nucleus
Electron-nucleus electrical attraction
Car cornering
Friction between types and the road
Moon revolving around the Earth
Moon-Earth gravitational attraction
Other satellites orbiting the Earth
Satellite-Earth gravitational attraction
Low Earth and Geostationary Orbits
A low Earth orbit is an orbit higher than 250km above the Earth’s surface and lower than about
1000km, having a revolution period of about 90 minutes at speeds of about 27000kmh-1. They are
used as weather, military and sky satellites, Earth monitoring and remote sensing satellites to
measure plant cover, surface water, chemical composition or subsurface ore bodies. This includes
the space shuttle and the Hubble Space Telescope.
A geostationary orbit is at an altitude where the period of the orbit matches the Earth’s rotation on
its axis, which corresponds to an altitude of about 35800km. If over the Equator, this orbit allows the
satellite to ‘park’ over a fixed point of the Earth, which is useful for communications satellites as
receiving dishes only need to point at a fixed point in the sky to maintain contact, but a quarter
second delay in bouncing a signal off the satellites exists due to their distance from Earth.
Space 4
Orbital Velocity
Orbital velocity is the instantaneous speed in the direction drawn as a tangent from a point on the
orbital path. The gravitational force between the satellite and the central body provides the
centripetal force.
𝐹𝑐 = 𝐹𝐺
𝑚𝑠 𝑣 2 𝐺𝑚𝐸 𝑚𝑠
=
𝑟
𝑟2
𝐺𝑚𝐸
𝑟
∴𝑣=√
, which is the orbital velocity
Thus, an object’s orbital velocity depends on the radius and mass of the central object and the
altitude above its surface. The central body’s mass and radius are fixed, so altitude is the only
variable determining the velocity required. The greater the radius of the orbit, the slower the orbital
speed required.
Orbital Decay
Low Earth Satellites orbit at altitudes of less than 1500km, while the exosphere extends out to
9600km, meaning there is some atmospheric drag from friction with gas molecules, resulting in a
loss in altitude and orbital decay. Atmospheric drag depends on the size of the satellite and the
density of the air along the orbit, which is affected by the time of day, season, latitude and
longitude.
However, solar wind and incoming solar radiation can heat up the outer atmosphere, causing it to
expand and increase in density and height, worsening orbital decay. As a satellite slows, it loses
altitude and begins a spiral downwards as it encounters higher density air and drag. By the time a
satellite is below 200km, it only has a few hours left before colliding with the Earth, with the heat of
re-entry causing some satellites to vaporise before impacting.
Re-entry
An orbiting spacecraft possesses a lot of kinetic and gravitational potential energy that causes
extreme heat when re-entering due to friction with the atmosphere. This could be reduced by using
an ablative heat shield (e.g. Apollo missions) or insulating surfaces (e.g. the Space Shuttle’s silica
tiles). The heat can be minimised by lengthening the time that the energy is converted to heat or
using blunt aircraft, which generate a shockwave of air in front of it to absorb the heat.
The deceleration of a re-entering spacecraft produces high g-forces. They can be better tolerated by
reclining the astronaut so the blood doesn’t flow away from the brain and by fully supporting the
body. They can also be minimised by extending the re-entry and slowing the rate of descent.
As heat builds up around the craft on re-entry, the atoms in the air around it ionise, forming a layer
around the spacecraft that radio signals cannot penetrate, preventing communication with the
ground. Space shuttles have 16 minutes of ionisation blackout on re-entry.
Reaching the surface and touching down softly on the surface can be achieved with parachutes and
landing in the ocean, while the space shuttle flies in a series of sharp-banking S turns to control its
descent and land on a runway after deploying speed brakes and landing gear.
The optimum angle for re-entry is 6.2°. If the angle is too shallow and under 5.2°, the spacecraft may
bounce off the atmosphere can become lost in space. If the angle is too steep and over 7.2°, the
spacecraft will burn up due to the heat of re-entry.
Space 5
Contribution to Space Exploration
Robert Goddard (1882-1945) was an American college professor in physics with a passion for
rocketry and was also an engineer and an experimentalist. He conducted practical experiments in
rocketry and determined that rockets worked better in a vacuum than in air, when most people
believed that air was necessary for the rocket to push against.
Goddard developed solid and liquid propellant rockets with the first successful flight with liquid
propellants in 1926. He built and tested the world’s first liquid-fuel rocket, which solved many
technical problems, such as fuel valving for throttle, start and stop, fuel injection, engine cooling and
ignition. He was the first to use gyroscopes and vanes for guidance, and to separate the payload
from the rocket in flight and return it to Earth.
Formulae for Projectile Motion
 𝑣𝑥 2 = 𝑢𝑥 2
 𝑣 = 𝑢 + 𝑎𝑡
 𝑣𝑦 2 = 𝑢𝑦 2 + 2𝑎∆𝑦
 ∆𝑥 = 𝑢𝑥 𝑡
1
 ∆𝑦 = 𝑢𝑦 𝑡 + 2 𝑎𝑡 2
Space 6
Part 3: The Solar System is held together by gravity
Effects of a Gravitational Field
A gravitational field is where a mass experiences a gravitational force. For smaller masses in the
field, the force of attraction to the massive object depends on their mass and the distance between
their centres. Therefore, the larger a mass and the closer it is to the Earth, or other large body, the
stronger the attraction
Newton’s Law of Universal Gravitation is that an object attracts every other object in the universe,
with factors determining the force of attraction being the mass of each of the objects and the
𝐺𝑚 𝑚
distances between their centres of mass. Mathematically, this is represented as 𝐹 = 𝑟12 2.
Gravitational attraction provides the centripetal force required for a satellite to orbit a planet.
𝐹𝑐 = 𝐹𝐺
𝑚𝑠 𝑣 2 𝐺𝑚𝐸 𝑚𝑠
=
𝑟
𝑟2
𝑣=√
𝐺𝑀
𝑟
and 𝑣 =
2𝜋𝑟
𝑇
This shows that a satellite’s required orbital velocity depends on its mass and orbital radius and
describes the relationship between the radius of an orbit and the period for any satellite.
The Slingshot Effect
The slingshot effect is performed to achieve an increase in speed or a change in direction. A space
probe is aimed close to a planet in a hyperbolic orbit, and is caught by its gravitational field in its
approach, being swung around the planet, but its gravity. The speed acquired is sufficient to throw
the space probe back out again away from the planet and the space probe will have sped up relative
to the Sun as it has acquired kinetic energy from the planet. However, from the planet, it appears as
though the space probe has returned to its initial velocity upon leaving the planet’s gravitational
field.
Space 7
Part 4: Current and emerging understanding about time and space
Earlier Models on the Transmission of Light
In the 19th century, it was concluded that light is a waveform. It was assumed that that light would
require a medium for propagation like other waves. However, the medium could not be found, so
aether was proposed as the medium. Its properties include: being transparent, being stationary and
filling all space, permeating all matter, having a low density and having high elasticity to propagate
light.
The Michelson-Morley Experiment aimed to find the Earth’s velocity through the aether, by finding
whether the speed of light varies with direction through the medium, through the use of
interference. In the experiment, they used an interferometer which had a half-silvered mirror and 2
mirrors to create 2 paths for the light to travel through and observed an interference pattern at a
screen.
If aether wind existed, then one light ray would be slower than the other, so when the apparatus is
rotated, the rays would be interposed and the interference pattern would shift. However, even
when the apparatus was rotated through 90°, there was no change in the interference pattern.
Experiments can be performed to test the predictions from theories, and from these results, the
validity of a theory can be determined. The Michelson-Morley experiment performed in 1887 tried
to test the prediction according to the aether model that an aether wind should exist. Although the
instruments satisfied requirements of sensitivity and did not measure an aether wind, the theory
was not abandoned and modifications were offered and later disproved.
In 1905, Einstein proposed the theory of relativity that did not require the aether model. This theory
had predictions to be tested, but was limited by the technology of the time. Thus, the MichelsonMorley experiment helped to show that the aether model was incorrect and prompted Einstein’s
development of the theory of relativity, which was later able to be proved true.
Frames of Reference
A frame of reference is a rigid framework from which relative changes can be measured. An inertial
frame of reference is moving at a constant velocity or is in a state of rest. A non-inertial frame of
reference is rotating or accelerating.
Relativity
Galileo proposed the principle of relativity that all steady motion is relative and cannot be detected
without reference to an exterior point, which is embodied in Newton’s First Law of Motion.
In 1905, Einstein published a paper proposing a Special Theory of Relativity, which presented 2
postulates, including that the laws of physics are the same in all inertial frames of reference and that
the speed of light is constant at the same value, c, which is independent of the observer’s velocity.
The Constancy of the Speed of Light
In Einstein’s thought experiment, the occupant of the train travelling at the speed of light will see
their reflection in a mirror, but an observer next to the train would see light travel at twice its
normal speed according to Newtonian physics, in which time is absolute, while distance and velocity
are relative. However, as the speed of light is assumed to be constant under Einstein’s theory, then
the observer will see the light travelling normally, but time and distance as perceived by the
𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒
occupant and the observer will be different to satisfy 𝑐 = 𝑡𝑖𝑚𝑒
Space 8
This means that the observer will see the light take twice as long to reach the mirror and it is
perceived that that time has slowed down in the train. As such, Einstein’s theory states that the
speed of light is absolute and that space and time are both relative quantities that depend on the
motion of the observer.
In 1793, the metre was defined by the French government as 1 × 10−7 times the length of the
Earth’s quadrant passing through Paris. The arc was surveyed and 3 platinum standards and iron
copies were made, but later, the survey was found to be incorrect. In 1875, when the Systeme
Internationale (SI) of units was established, the metre was defined as the distance between 2 lines
scribed on a bar of platinum-iridium alloy.
The current definition of the metre uses the constancy of the speed of light in a vacuum and the
accuracy of the definition of one second to achieve an accurate and accessible result. A metre is
1
defined as the length of the path travelled by light in a vacuum during 299 792 458 of a second.
Consequences of Special Relativity
The relativity of simultaneity
If an observer judges 2 events to be simultaneous, they generally will not be judged as simultaneous
by an observer in a different frame of reference. Einstein’s thought experiment was that a light
operated by an occupant is fitted in the centre of a train with light operated doors. The occupant
and exterior observer would see the doors opening at different times.
The equivalence between mass and energy
A rest mass is equivalent to a certain amount of energy. Under certain circumstances, a mass can be
converted into energy and vice versa. Some mass is converted into energy in nuclear fission
reactions, while the collision of matter and anti-matter results in the complete conversion into
energy. 𝐸 = 𝑚𝑐 2
Length contraction
The length of an object in its rest frame is the proper length, 𝐿𝑜 , but observers in different frames of
reference in relative motion will perceive length, 𝐿𝑣 , to be shorter
Time dilation
The time taken for an event to occur in its rest frame is the proper time, 𝑇𝑜 , but observers in
different frames of reference in relative motion will perceive time, 𝑇𝑣 , to be longer
Mass dilation
Special Relativity results in the increase in mass of a moving object as its velocity increases. This
effect is only noticeable at relativistic speeds, so as an object is accelerated close to the speed of
light, its mass increases and more energy needed to produce the same acceleration, making further
accelerations more difficult.
Implications of Special Relativity on Space Travel
If relativistic speeds can be reached, the nearest stars could be reached in several years, but due to
time dilation and length contraction, the journey would take significantly less time, but accelerating
to relativistic speeds would incur considerable energy costs due to the conversion of energy into
mass.
Space 9
From the Earth, the clocks on the spacecraft are moving slowly, so less time passes on the spacecraft
than on the Earth. The occupants of the spacecraft perceive that the length of the journey has
contracted to a significantly shorter distance which they cover in less time.
Evidence Supporting Special Relativity
A theory requires supporting experimental evidence before it is adopted. When Einstein proposed
his special theory of relativity in 1905 and his general theory of relativity in 1915, there was a lack of
experimental evidence and technological capacities to verify the predictions. Einstein used thought
experiments to investigate situations that could not be tested in reality, which was due to
limitations in the technology of the time. As technology improved in the 20th century, relativity
theory predictions became testable.
The existence of time dilation has been experimentally verified by comparing atomic clocks that
have been flown over long journeys with clocks that had remained stationary over the same period,
with the clock in the plane lagging slightly behind. Special relativity is also supported by the energy
yield from converted mass in nuclear reactions that show the equivalence of energy and mass. Also,
when particles are accelerated to near-light speeds in particle accelerators, there is an observed
increase in their mass.
Another piece of supporting evidence involves muons, which is a subatomic particle created in the
upper atmosphere by cosmic rays. They are unstable and disintegrate in a lifetime of 2.2µs. In this
time, they should be unable to reach the Earth’s surface despite their speeds of ~0.999c. However,
they are found on the Earth’s surface, as relative to observers on the Earth, their lifetime is dilated
to 49.2 µs, which is enough for them to reach Earth.
Formulae for Special Relativity
 𝐸 = 𝑚𝑐 2
𝑣2

𝑙𝑣 = 𝑙𝑜 √1 − 𝑐 2

𝑡𝑣 =
𝑡𝑜
2
√1−𝑣2
𝑐

𝑚𝑣 =
𝑚𝑜
2
√1−𝑣2
𝑐