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Knowledge and understanding
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When you have finished this chapter, you should be able to:
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explain the concept of escape velocity in terms of the:
–
gravitational constant
–
mass and radius of the planet
outline Newton’s concept of escape velocity
identify why the term ‘g-forces’ is used to explain the forces acting
on an astronaut during launch
discuss the effect of the Earth’s orbital motion and its rotational
motion on the launch of a rocket
identify that a slingshot effect can be provided by planets for space
probes
analyse the changing acceleration of a rocket during launch in
terms of the:
–
Law of Conservation of Momentum
–
forces experienced by astronauts
analyse the forces involved in uniform circular motion for a range of
objects, including satellites orbiting the Earth
compare qualitatively low-Earth and geostationary orbits
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•
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•
•
•
•
•
3
Spaceflight
define the term ‘orbital velocity’ and the quantitative and qualitative
relationship between orbital velocity, the gravitational constant,
mass of the central body, mass of the satellite and the radius of the
orbit using Kepler’s Law of Periods
account for the orbital decay of satellites in low-Earth orbit
investigate the contribution to the development of space
exploration of Tsiolkovsky, Oberth, Goddard and von Braun.
solve problems and analyse information to calculate the centripetal
force acting on a satellite undergoing uniform circular motion about
mv 2
the Earth using F ____
r
GM
r3
___
solve and analyse problems using __
T2
42
discuss issues associated with safe re-entry into the Earth’s
atmosphere and landing on the Earth’s surface
identify that there is an optimum angle for safe re-entry into the
Earth’s atmosphere and the consequences of failing to achieve this
angle.
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3.1 Escape velocity
In 1867, Isaac Newton published a universal theory to calculate the orbits
of the planets and describe the effect of gravitation within the Universe.
You will study that theory in detail in Chapter 4. In this theory, he
postulated that it was the same force that keeps the planets in orbit around
the Sun, and the Moon in orbit around the Earth as well as causing objects
to drop to the surface of Earth – gravity.
Galileo knew that when a cannon is fired, the ball will be pulled by
gravity and ‘fall’ towards the centre of the Earth. However, as you know,
the ball is subject to a forward motion as well, so it moves in a horizontal
motion as well as a vertical motion. This forces the cannonball to fly in a
curved line of motion. When the cannon uses more gunpowder, then the
ball will be imparted a higher horizontal velocity and so will fall farther
away from its starting point.
Newton postulated that when the starting velocity of the ball is high
enough, the ball will never fall back to Earth but will go into orbit around
the Earth. His conclusion from this postulation was that the Moon is
constantly falling to Earth, which results in its elliptical orbit. He further
reasoned that this starting velocity, whereby a projectile would remain in
orbit, could be calculated using:
______
v
Figure 3.1 A satellite in orbit.
where
2Gm
rp
______p
v escape velocity
G is the universal gravitational constant 6.67 1011 N m2 kg2
mp mass of the planet
rp radius of the planet
Figure 3.2 NASA’s space shuttle blasts off.
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Chapter 3 \\ Spaceflight
This formula is found by combining our knowledge of kinetic energy
and gravitational energy, as escape velocity can be defined as the speed
where the kinetic energy of an object is equal to the magnitude of its
gravitational potential energy.
31
A
B
E
KE Ep
Gm m
rp
p
1mv 2 _______
__
2
2Gmp
v 2 ______
r
C
p
______
v
D
2Gm
rp
______p
The escape velocity is commonly described as the speed needed for an
object to ‘break free’ from a planet’s gravitational field, without any
additional speed or acceleration being given to it, so that it will never be
pulled back to the surface of that planet.
\\ WORKED EXAMPLE
Figure 3.3 A ball projected horizontally at increasingly
faster speeds will eventually end up in orbit.
escape velocity
The initial velocity that needs to
be imparted to a mass in order for
it to ‘break free’ from a planet’s
gravitational field
Question 1
Calculate the escape velocity for the Earth given:
mE 5.974 1024 kg
rE 6.378 106 m
G 6.67 1011 N m2 kg2
Answer
Substituting into the escape velocity formula gives:
_____
v
2GmE
_____
rE
________________________
2(6.67 10 ) (5.974 10 )
________________________
11
24
6.378 10
6
11 200 m s1
40 200 km h1
The escape velocity for Earth is of the order of 40 000 km h1! Note that it is
independent of the mass of the object being launched.
Newton had no way to test his idea, but we now know that in practice
speeds this high cannot be achieved on Earth. The friction and heat that
such a speed would cause would mean that the object would burn up
shortly after being fired. Note also that escape velocity is independent of
the mass of the escaping object. It does not matter if the mass is 1 kg or
100 kg – the escape velocity from the same point in the same gravitational
field is always the same. What differs is the amount of energy needed to
accelerate the mass to achieve the escape velocity.
3.2 g-forces
g-force
g-force is a measurement of an object’s acceleration expressed in
multiples of the gravitational acceleration at the Earth’s surface or g.
Astronauts in a rocket at rest on the launch pad will experience a 1g
A measurement of an object’s
acceleration expressed in
multiples of gravity or g
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force equivalent to the product of their mass and gravity (F mg).
g-forces are experienced by astronauts, fighter pilots and riders on
a roller-coaster. When the roller-coaster is moving upwards, the
passengers typically feel as though they weigh more than usual as the
upwards acceleration induces a higher positive g-force. As the riders
move downwards, they often feel a negative acceleration that makes
them feel lighter than usual or even ‘weightless’.
To calculate g-force, we need to take into account that the body is
already influenced by a force of 1g towards the Earth. For example:
•
•
•
•
An object at rest with respect to the Earth experiences a g-force of
0g 1g, or just 1g (‘normal weight’).
An object in free-fall (accelerating downwards at 1g relative to the
Earth) experiences a g-force of 1g 1g 0g (‘weightless’)
An object accelerating upwards at 1g relative to the Earth experiences
a g-force of 1g 1g 2g (‘twice normal weight’).
An object accelerating downwards at 2g relative to the Earth
experiences a g-force of 2g 1g 1g (‘negative g’).
(b)
(a)
(c)
a0
g
g-force g
g
a g (down)
g
g-force 0
a g (up)
g
g
g-force 2g
Figure 3.4 g-force acting on (a) object at rest, (b) falling object and (c) upward accelerating object.
When dealing with rockets, the net upwards
force, ma, is the resultant of upwards thrust,
T, and downwards gravity.
i.e. ma T mg
The above formula becomes:
(T mg) mg
g-force ____________
mg
which can be simplified to:
T
g-force ___
mg
When on the Earth’s surface, this can be written as:
apparent acceleration a g
or
apparent weight ma mg
In which case:
apparent weight
g-force _______________
true weight
ma mg
_________
mg
Note: In this situation, g is considered to be a positive () force.
\\ WORKED EXAMPLE
Question 2
A rocket and its 200 kg fuel load have a total mass of 600 kg and develop a
thrust of 8000 N. Calculate the maximum g-force that would be experienced by
its astronauts.
Answer
Data: m 600 kg; fuel 200 kg; F ma 8000 N; g-force ?
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Maximum acceleration is experienced just prior to the fuel being exhausted, at
which point the rocket has a mass of 400 kg.
T
g-force ___
mg
8000
________
(400)(9.8)
2.0 g
The maximum acceleration experienced by the astronauts is 2.0g.
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Human g-force tolerances
Human tolerances to g-forces depend on the magnitude of the g-force,
the length of time over which it is applied, the direction in which it acts,
the location of the application and the posture of the body.
Aircraft in particular exert a g-force along the vertical axis of the spine.
g-forces are often directed towards the feet, which causes a loss of blood
from the head. This causes problems with the eyes and brain, so that as
the g-force increases, vision loses hue, then tunnel vision will occur until
at still higher g, complete loss of vision can occur while still conscious.
Past this point of ‘blacking-out’, loss of consciousness may be experienced
and, while individual tolerances vary, generally a person can experience
about 5g before this occurs.
The human body is considerably better at experiencing g-forces that
are perpendicular to the spine. For this reason, astronauts are propelled
into space lying down on a body-contoured ‘seat’ for maximum body support.
Early astronauts were subjected to launch and re-entry forces of up to 6g and
more. In the space shuttle, only a little more than 3g is experienced on lift-off.
Pilots and astronauts also wear g-suits – pressurised suits that prevent their
blood pooling too much under severe g-forces.
g-force blackout
3.3 Rotation of the Earth and
rocket launch
A rocket launch site is built as far away as possible from populated areas
to prevent risk to bystanders in the event of a catastrophic failure. For this
reason, many launch sites are built close to major bodies of water. The
best-known launch sites are Cape Canaveral in Florida, Vandenberg Air
Force Base in California and Russia’s Baikonur in Kazakhstan. Launch
sites are usually constructed as close to the Equator as possible to allow
rockets launching eastbound to receive extra velocity from the Earth’s
rotation. The Earth’s rotational speed is about 1700 km h1 and it is itself
travelling at about 107 000 km h1 as it orbits the Sun. These speeds can be
utilised when launching rockets from the Earth’s surface, particularly near
the Equator, and enable the rocket speed to be ‘boosted’ because of the
Earth’s motion, as shown in Figure 3.5. The timing of the launch is
important to enable the rocket to gain speed from the rotation of Earth,
and to then join in with the orbit around the Sun and thus gain speed
from it. This is known as the slingshot effect and its use saves fuel, time
and expense.
Sun
motion
of the Earth
Figure 3.5 The slingshot effect.
slingshot effect
Launching a rocket so as to make
use of both the spin of the Earth
and the orbital speed of the Earth
around the Sun to enable it to
reach must faster speeds
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3.4 Changing rocket
acceleration
multistage rocket
Stage thrust (N × 106)
A rocket using two or more stages,
each of which contains its own
engines and propellants. As the
fuel from one stage is exhausted,
that stage is jettisoned and the
next stage fires up
Unlike aeroplanes, rockets are often multistaged, meaning that they use
two or more stages, each of which contains its own engines and
propellants. The result is effectively two or more rockets stacked on top
of or attached next to each other. Stages are jettisoned when they run out
of fuel, thus reducing the mass of the remaining rocket and enabling
greater speeds to be attained. As the mass of fuel reduces through a
stage’s life, the acceleration of the rocket increases, gradually increasing
to a maximum just prior to the stage running out of fuel and being
jettisoned. At the point of jettisoning, the rocket coasts along until the
second stage rocket fires up and quickly develops the force needed for
the rocket to again be accelerating. The mass of a rocket is generally
about 90% fuel as such large amounts of thrust are required to gain the
acceleration and speed needed to escape the Earth’s gravitational field.
42
40
38
36
34
32
30
0
20
40
60
80
100 120 140 160
Range time (s)
Figure 3.6 The second stage of a Saturn V rocket
being lowered onto the first stage.
Figure 3.7 Saturn V performance thrust. Note the increase in thrust as fuels are burnt and mass of
rocket decreases.
3.5 Uniform circular motion
Calculations involving
centripetal forces and circular
motion, pages 19–20, Practical
Physics for Senior Students, HSC
v
v
v
Figure 3.8 The velocity of an object moving in
a circle is at a tangent to the circle.
An Earth satellite moving at constant speed in orbit around the Earth
undergoing circular motion may be considered to be an example of
uniform circular motion. Even though the speed of the satellite is constant,
its direction is constantly changing, and hence its velocity is changing. In
other words, for a satellite to be undergoing uniform circular motion
means that its speed is constant but not its velocity. The velocity is
constantly being changed by a force perpendicular to the direction of
motion and directed to the centre of the circle.
It can be shown mathematically that the centripetal acceleration acting
towards the centre of the circle for an object undergoing uniform circular
motion is:
v2
ac __
r
From Newton’s Second Law (F ma) we can also see that:
mv 2
F ____
r
where F centripetal force (N)
m mass of object (kg)
v velocity of object (m s1)
r radius of circle (m)
ac centripetal acceleration (m s2)
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35
\\ WORKED EXAMPLE
Question 3
Circular motion
A 300 kg satellite is orbiting the Earth at an altitude of 250 km. If its orbital
speed is 27900 km h1, find the centripetal force acting on it and its centripetal
acceleration. Assume the radius of the Earth to be 6380 km.
Answer
First change the orbital speed into m s1:
speed in km h1
speed in m s1 _____________
3.6
27900
_____
3.6
7750 m s1
Calculate the orbital radius:
orbital radius 6380 250
6.63 106 m
Data: m 300 kg; v 8333 m s1; r 6.63 106 m; F ?
mv 2
F ____
r
300 77502
___________
6.63 106
2718 N
So the centripetal force is 3142 N towards the centre of the Earth.
v 2 __
F
ac __
r
m
2718
____
300
9.06 m s2
The centripetal acceleration is 10.5 m s2 towards the centre of the Earth.
Consider also the period of such an orbiting satellite. The distance of
one circumference of the orbit is given by 2r, and since the period T is
the time for one complete revolution:
distance
v ________
time
2r
____
T
2r
____
So T v .
3.6 Low-Earth and
geostationary orbits
There are two orbital altitudes that satellites will be placed in around the
Earth – low-Earth orbits and geostationary orbits.
Low-Earth orbits lie above the top of the atmosphere to avoid
atmospheric drag, and below the van Allen radiation belts, to avoid
interference from the high levels of radiation. This means that they are
found between 250 and 1000 km above the surface of the Earth. Satellites
in low orbit need to travel at very high speeds to maintain stability in their
low-Earth orbit
Orbit that lies above the top of
the atmosphere and below the van
Allen radiation belts. They are
found between 250 and 1000 km
above the surface of the Earth
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geostationary orbit
Orbit that lies at about 42 000 km
above the surface of the Earth.
Satellites in geostationary orbit
maintain a constant position
relative to the surface of the
Earth
orbital velocity
The velocity required by a
satellite or spacecraft to enter
and maintain a particular orbit
around the Earth or some other
celestial body
orbit, completing one revolution of the Earth in about 90 minutes. A
satellite in orbit at 500 km above the surface of the Earth would need to
orbit with a speed of 27 000 km h1 to remain in a stable orbit. Low-Earth
orbit satellite systems require several dozen satellites to provide coverage
of the entire planet. Although they are above the atmosphere, low-Earth
satellites are still subject to a small amount of drag which will over time
rob the satellite of energy and result in decay of its orbit. The size of the
drag on the satellite relates to the density of the atmosphere in the
satellite’s orbit. This can be affected by increased solar activity, which
heats and expands the upper atmosphere. It is also affected by the seasons,
time of day, latitude and longitude. As the satellite loses energy, its altitude
decreases, bringing it into denser atmosphere. As the satellite loses more
energy, a downward spiral of decay commences until it becomes so hot
near the atmosphere of the Earth (about 200 km) that it burns up.
Geostationary orbits lie at about 42 000 km above the surface of the
Earth. Satellites in geostationary orbit maintain a constant position
relative to the surface of the Earth. Such satellites are used for
communication, weather observation and military ‘spy’ satellites. The
orbit is directly above the Equator and the period is 23 hours and
56 minutes. A single geostationary satellite will provide coverage over
about 40% of the planet. Because they circle the Earth at the Equator, they
are not able to provide coverage at the northernmost and southernmost
latitudes.
Orbital velocity is the velocity required by a satellite or spacecraft to
enter and maintain a particular orbit around the Earth or some other
celestial body. It is dependent on the mass of the planet and the distance
from the centre of that planet. It is independent of the mass of the satellite.
Mathematically, the particular velocity that a satellite will need to stay in
orbit is expressed as:
_____
v
where
Gm
rp
_____p
v the velocity
G the universal gravitational constant
mp the mass of the planet
rp the radius of orbit from planet’s centre
\\ WORKED EXAMPLE
Question 4
Calculate the orbital velocity required by satellites at altitudes:
a 250 km
b 500 km
Consider the radius of the Earth to be 6.38 106 m and the mass to be
5.97 1024 kg.
Answer
a
At an altitude of 250 km:
____
v
Gm
rp
____p
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______________________
(6.67 10 )(5.97 10 )
______________________
(6.38 10 250 10 )
11
24
6
3
7750 m s1
27 900 km h1
b
The velocity is 7750 m s1 or 27 900 km h1.
At an altitude of 500 km:
____
v
Gm
rp
____p
______________________
(6.67 10 )(5.97 10 )
______________________
11
24
(6.38 106 500 103)
7608 m s1
27 390 km h1
The velocity is 7608 m s1 or 27 390 km h1.
Question 5
Calculate the periods of the above satellites.
Answer
Kepler’s Second Law
a
At an altitude of 250 km:
2r
T ____
v
2(6.38 106 250 103)
_______________________
7750
5375 s
89.6 min
b
The period is 89.6 min.
At an altitude of 500 km:
2r
T ____
v
2(6.38 106 500 103)
_______________________
7608
5682 s
94.7 min
The period is 94.7 min.
Solving problems using Newton’s
modification of Kepler’s Third
Law, pages 20–21, Practical
Physics for Senior Students,
HSC
3.7 Kepler’s Law of Periods
Johannes Kepler, working with data painstakingly collected by Tycho
Brahe without the aid of a telescope, developed three laws which described
the motion of the planets across the sky.
1 The Law of Orbits – All planets move in elliptical orbits, with the Sun
at one focus.
2 The Law of Areas – A line that connects a planet to the Sun sweeps out
equal areas in equal times (see Figure 3.9).
3 The Law of Periods – The square of the period of any planet is
proportional to the cube of the radius of its orbit.
Sun
T
1
2
T
Area 1 = Area 2
Figure 3.9 Kepler’s Second Law of Orbital Motion.
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It is this third law that we are interested in – the Law of Periods.
Expressed mathematically, Kepler’s Law says:
3
GM
r ____
___
T2
where
orbital ratio
The ratio between the cube of the
orbital radius and the square of
r 3 . For any
the period of orbit or __
T2
given planet the ratio is constant
regardless of the altitude of orbit
4 2
r the orbital radius
T the orbital period
G the universal constant of gravitation
M the mass of the planet
This says that for any given planet, the ratio between r 3 and T 2 is
constant. So it does not matter what the altitude of the orbit is, the
r3
relationship __
is a constant for that planet. This is known as the orbital
T2
ratio.
\\ WORKED EXAMPLE
Question 6
Kepler’s third law
Kepler’s third law can be established quite
easily given our knowledge of both
centripetal and gravitational force.
Centripetal force can be rewritten as
follows:
Fr
mv 2 ⇒ v 2 __
F ___
m
r
where m is the mass of the satellite, r is the
orbital radius and v is the orbital velocity.
Substituting Newton’s universal law of
GMm ) gives
gravitation (F _____
r2
GM
GMm
r
2
_____
___
v 2 __
m r
r
where M is the mass of the planet.
circumference
distance __________
Now since v ______
time
period
2 2
2r, v 2 _____
4 r
___
T
T2
We can join these two equations for v 2
together:
GM
4 2r 2 ___
_____
r
T2
2
2
4 __
T
____
r3
GM
And thus we have Kepler’s third law.
Calculate the orbital ratio for an Earth satellite. Assume the radius of the Earth
to be 6.36 106 m and its mass to be 5.97 1024 kg.
Answer
GM
r 3 ____
T 2 4 2
____
(6.67 1011)(5.97 1024)
______________________
4 2
1.01 1013
The orbital ratio for Earth is 1.01 1013.
Note that this means that this is the ratio for a satellite in orbit at 250 km and
one at 42 000 km.
Question 7
Two planets, A and B, travel around a star in circular orbits in the same
direction (Figure 3.10). Planet A has a radius of r from the centre of the star and
planet B has a radius of 4r. Planet A completes one revolution about the star in
time T. Calculate the period of planet B.
B
4r
A
r
Figure 3.10
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Answer
Because the planets are revolving around the same gravitational mass, the star,
the ratio of their orbital periods will be the same. Thus:
r A3 ___
rB 3
___
T A2 T B2
r A3 _____
(4r )3
substituting in ___
A2
2
TA
TB
gives T B2 r 3 64r 3 T 2
_____
T B 64T 2
8T
The period of planet B is 8T.
3.8 Re-entry
Probably the most difficult manoeuvre during a space flight is the re-entry
into the Earth’s atmosphere. If the atmosphere is approached at too small
an angle, the craft is likely to bounce off the atmosphere and continue
journeying in space. If, on the other hand, the angle is too great, then the
heat associated with re-entry becomes too great and it is very difficult to
insulate a spacecraft from the heat caused by the friction of the atmosphere.
The heat of re-entry usually burns up meteors and uncontrolled spacejunk (decayed orbit satellites) where temperatures of up to 2500°C are
common for controlled entry, and much higher for uncontrolled entry.
For the Apollo missions, the angle of re-entry was between 5.2° and
7.2° – a very small window to aim for.
A spacecraft has a lot of kinetic and potential energy, which it needs to
lose to be able to safely land. To be in a stable Earth orbit in the first place,
the craft must have attained and maintained a critical velocity of around
30 times the speed of sound – 13 km s1! There are two ways that a
spacecraft or satellite can deal with the heat of re-entry – ablative
technology and insulating tile technology. In ablative technology, the
surface of the heat shield melts and vaporises, carrying away heat in
the process. This is the technology that protected the Apollo spacecraft.
The space shuttles use insulating tile technology and are protected by
utilising special silica tiles. Silica (SiO2) is an extremely good insulator – it
is possible to hold the edge of a space shuttle tile and at the same time
heat up the centre of the tile with a blowtorch! The tile insulates so well
that no heat makes it out to the edges. The kinetic and potential energies
are dissipated as heat and the black tiles on the bottom of the space
shuttle act as the main heat shield as the spacecraft makes its fiery hourlong descent.
Another problem of re-entry for astronauts is the large g-forces that
they may experience. Even with the rockets retro-firing, there are
enormous amounts of both kinetic and potential energy to be dissipated
and large decelerations. To overcome the g-forces, the astronauts need to
once again be in a reclining position, backwards to the direction of travel,
to prevent the blood rushing to their feet.
Re-entry is made difficult by the communication blackout that occurs
due to the high heat, which ionises the oxygen and nitrogen in the
atmosphere and creates an ionised blanket around the spacecraft,
preventing communication for about 15–20 minutes.
re-entry
The return of a spacecraft into
the Earth’s atmosphere including
its descent to Earth
ablative technology
Protection of the landing vessel
from the heat of re-entry by the
vaporisation of the protective
tiles
insulating tile technology
Protection of the landing vessel
from the heat of re-entry by
highly insulating tiles
Spaceshuttle re-entry
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Rocket pioneers
Figure 3.11 Robert Goddard holding the launch frame
of ‘Nell’, the first liquid-fuelled rocket.
The early rocket scientists
who contributed to space
flight, pages 17–18, Practical
Physics for Senior Students,
HSC
For centuries, people have dreamed of space travel, but the 20th century
saw the dreams change into reality. Rockets were, for the first time,
recognised as the only means of practical travel into space, and design
advances transformed them from oversized fireworks to the space-going
machines such as the Apollo missions and the space shuttles.
Jules Verne was a prolific science fiction author in the 19th century
and, along with H.G. Wells, was an inspiration and a source of wonder to
many would-be rocket designers.
Konstantin Tsiolkovsky (1857–1935) grew up in Russia and, probably
more than any other person, did much to make space travel a reality.
Although he did not actually launch any rockets, he developed many of
the principles and techniques still used in rocketry today, although it was
only in his old age that he received recognition for his ideas – ideas for
which he is considered to be the father of human spaceflight. His most
important work, published in 1903, calculated the horizontal speed of
11 km s1 for a minimal orbit around the Earth. Although much of his work
was theoretical, he did experiments to show that living creatures could
survive accelerations of up to 60 m s1 (6g), but not much more. He
further suggested that space travel could be achieved and controlled by
means of a multistage rocket fuelled by liquid oxygen and liquid hydrogen.
Although Tsiolkovsky is rightly known as the founder of modern rocketry,
his ideas were not known outside his home country of Russia and so were
not familiar to his Western contemporaries such as Robert Goddard
(1882–1945). Goddard was an American physics lecturer who pioneered the
field of controlled liquid-fuelled rocketry. He is best remembered for launching
the world’s first liquid-fuelled rocket on 16 March 1926. During his life, he
patented several multistage rockets, which would become important
milestones in the history of rocketry, and he patented over 200 rocketry ideas.
Goddard had a long career in rocket development, pioneering many ideas
and techniques, and launching many rockets, that were to become the basis
of spaceflight in the future. Much of his work was completed in secrecy, due to
early ridicule of his work, and so it is only in hindsight that his real contribution
to rocketry can be appreciated. As well as launching the first liquid-fuelled
rocket, he sent the first payload (a camera and barometer) up in a rocket (and
brought it back to Earth by parachute), he first used vanes in the rocket motor
blast for directional guidance and he first developed pumps suitable for rocket
fuels which improved the efficiency of the rockets from 2% to 64%.
Hermann Oberth (1894–1989) is also considered to be one of the
founding fathers of rocketry and spaceflight. Although Tsiolkovsky,
Goddard and Oberth were never active collaborators, their achievements
were parallel in many ways. Between World Wars I and II, especially in the
1930s, rocket clubs and their enthusiasts were very active in Germany,
Russia and the United States. Oberth was a German theorist who actively
promoted the idea of spaceflight through his self-published doctoral
thesis. He was considered to be the foremost authority on rocketry outside
the United States. Oberth was enlisted in 1929 by German film director
Fritz Lang to act as a consultant on what was to be the first film on space
travel – ‘The Woman in the Moon’ – and many of his ideas were utilised in
the film’s making. He was also persuaded to build Germany’s first liquidfuelled rocket for the launch of the film, but an oxygen explosion, resulting
in the loss of one eye, saw the project shelved.
While a member of the German Society for Space Travel, a young
engineering student, Wernher Von Braun, assisted Oberth. Von Braun
went on to be the rocket engineer responsible for the V-2 rocket launched
by the German army and used to bomb London during World War II. Postwar, Von Braun became an American citizen and the most important
figure in putting the first Americans into space.
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Module 1
Chapter 3 \\ Spaceflight
41
Summary of spaceflight
•
•
Escape velocity is the initial velocity that needs to be imparted to a mass in order for it to
‘break free’ from a planet’s gravitational field.
Newton’s concept of escape velocity was that if a horizontally fired projectile could be fired
at high enough velocity, its projectile path would
result in it orbiting the Earth.
____
•
Escape velocity is given by the formula v •
•
•
•
•
•
•
•
•
.
Escape velocity is the speed where kinetic energy is equal to gravitational potential energy.
2Gm
____p
rp
g-force is a measure of an object’s acceleration expressed in multiples of gravity or g.
apparent weight
g-force is given by the formula g-force ______________.
true weight
Human tolerances to g-forces depend on the magnitude of the g-force, the length of time
over which it is applied, the direction in which it acts, the location of the application and the
posture of the body.
Fighter pilots wear special pressurised g-suits to help them deal with g-forces.
Rocket launch sites are built far away from populated areas and as close to the Equator as
possible to make use of the slingshot effect.
The slingshot effect refers to launching a rocket so as to make use of both the spin of the
Earth and the orbital speed of the Earth around the Sun to enable it to reach much faster
speeds.
Modern rockets are multistaged to enable greater speeds to be attained.
Satellites in orbit around a planet may be considered to be undergoing uniform circular
motion.
•
mv
Centripetal force acts towards the centre of a circle and is given by the formula F ___
r .
•
v
Using Newton’s Second Law (F ma), centripetal acceleration is given by ac __
r.
•
•
2r
The period of an orbiting satellite in circular motion is given by the formula T ___
v .
Low-Earth orbits lie above the atmosphere and below the van Allen radiation belts
(250–1000 km).
Satellites in orbit need to travel at very high speeds to maintain a stable orbit – around
30 000 km h1 – and take about 90 minutes to orbit the Earth.
Geostationary orbits lie at about 42 000 km above the surface of the Earth and maintain a
constant position relative to the surface of the Earth above the equator.
•
•
2
2
____
Gm
___p .
rp
•
The velocity that a satellite needs to stay in orbit is given by the formula v •
Kepler’s Law of Periods states that the square of the period of any planet is proportional to
the cube of the radius of the orbit:
GM
r ____
___
3
T2
•
•
•
•
•
42
r
k for some constant k particular to
A simpler way of using Kepler’s Law of Periods is __
T2
each planet.
The most difficult manoeuvre in spaceflight is re-entry into the Earth’s atmosphere a very
specific angle of entry is required to prevent the spacecraft either burning up or bouncing
off the atmosphere layer.
The two ways of dealing with the enormous heat of re-entry are ablative technology (the
surface of the heat shield melts and vaporises) and insulating tile technology.
Re-entry is made difficult not only by the angle of re-entry, but also by the g-forces experienced
by the astronauts and the communication blackout that occurs in the upper atmosphere.
The three non-collaborative ‘fathers’ of modern rocketry are the Russian Tsiolkovsky, the
American Goddard and the German Oberth.
3
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Nelson Physics
42
Stage 6 HSC
Review questions
Question 1
Explain Newton’s concept of escape velocity. Use a diagram to aid your answer.
Question 2
Calculate the escape velocity for the Earth, given the following information:
mE 6.0 1024 kg
rE 6.38 106 m
G 6.67 1011 N m2 kg2
Question 3
Calculate the escape velocity for a planet of mass 9.5 1026 kg and radius
3.4 108 m.
Question 4
Explain why escape velocities are neither practical nor possible in reality.
Question 5
Explain the meaning of the term ‘g-force’.
Question 6
Calculate the g-force acting on an object accelerating upwards at 2g relative to the
Earth.
Question 7
A 700 kg rocket with a 300 kg fuel load develops a thrust of 12 000 N. Calculate the
maximum g-force that would be experienced by its astronauts.
Question 8
Explain why g-forces pose serious problems for astronauts and fighter pilots. Explain
ways that these problems may be overcome.
Question 9
Explain with a diagram the meaning of the term ‘slingshot effect’. Describe why this
effect is utilised in rocket launches.
Question 10
Explain what is meant by the term ‘multistage’ rockets and justify their usage.
Question 11
A 500 kg satellite is orbiting Earth at an altitude of 340 km with an orbital speed of
28 000 km h1. Find the centripetal force acting on it and its centripetal acceleration.
Assume the radius of the Earth to be 6380 km.
Question 12
Calculate the period of a satellite orbiting Earth at a height of 4200 km.
Question 13
A rocket has a mass of 15 000 kg, of which 75% is fuel. If it develops a thrust of
200 000 N, calculate the g-force and rate of acceleration:
a
at lift-off
b
just prior to exhaustion of the fuel and jettisoning of the stage.
Question 14
Compare the orbiting velocities of two satellites orbiting Earth, one at an altitude of
300 km and the other at an altitude of 3000 km.
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Module 1
Chapter 3 \\ Spaceflight
43
Question 15
Calculate the altitude of an Earth satellite with a period of 15 hours.
Question 16
Calculate the period of a satellite that orbits the Earth with a radius of 20 000 km.
Question 17
Define the term ‘orbital decay’ in low-Earth orbits.
Question 18
Compare the differences between low-Earth and geostationary orbits.
Question 19
Explain why satellites in low-Earth orbits will eventually fall to Earth.
Question 20
Discuss two issues that must be considered for the safe re-entry into the Earth’s
atmosphere and subsequent safe landing on Earth.
Question 21
Calculate the orbital ratio for a satellite orbiting a planet, given that the radius of the
planet is 6.5 107 m and its mass is 1.2 1025 kg.
Question 22
Two planets X and Y travel around a star in the same direction, in circular orbits.
Planet X has a radius of r from the centre of the star and planet Y has a radius of
3r. Planet X completes one revolution about the star in time T. Calculate the period of
planet Y.
Question 23
Two planets A and B travel around a star in the same direction, in circular orbits.
Planet A completes one revolution about the star in time T. The ratio of the orbits of A
and B is 1:4. How many revolutions does planet B make about the star in the same
time T ?
Question 24
Describe in detail the contribution made to the development of space exploration by
either Tsiolkovsky, Goddard or Oberth. Use more resources than just this text to detail
your answer and include a correct bibliography of those resources.
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