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64 CHAPTER
Key Ideas3
Stage 2 Physics
3.3 Application Weather & Communication Satellites
The U.S.S.R. launched the first satellite, Sputnik 1, on October 4, 1957. Since then man has
launched many thousands of satellites into orbit around the Earth. These satellites have a wide
variety of purposes and objectives. Among these are:
š
Weather satellites which observe, photograph and analyse weather patterns.
š
Communication satellites which provide communication links between widely separated areas
and to remote communities.
š
Surveillance satellites which observe areas of the Earth’s surface for environmental, defence,
military and agricultural purposes.
š
Global positioning satellites which enable aircraft, ships and any person with the appropriate
equipment to precisely locate their position on the Earth by comparing the signals from several
satellites.
3.3.1 Possible Orbits
A satellite in a circular orbit is moving with uniform circular motion.
š
Therefore, there is a centripetal acceleration acting towards the centre of the circle, and so
there must be a force that causes this centripetal acceleration. The direction of this force must
also be towards the centre of the circle.
š
But this force is the force of gravitational attraction. The gravitational force acts along the
line joining the centres of the two bodies, and so the force on the satellite is directed towards
the centre of the Earth.
The implication of the above two points is that the only circular orbits possible are those such
that their centre coincides with the centre of the Earth.
Figure 3.4(a) illustrates some possible orbits, while Figure 3.4(b) illustrates two satellite orbits that
are not possible.
Fig. 3.4 (a)
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Essentials Text Book
Fig. 3.4 (b)
Gravitation GRAVITATION
and Satellites AND SATELLITES 65
3.3.2 Geostationary Satellites
A geostationary satellite is one that is always above the same point on the Earth’s surface.
You will find them sometimes referred to as geosynchronous orbits.
These satellites are extremely useful for communication purposes, for the satellite is always in the
same position relative to the Earth, and this simplifies communication in the following ways.
š
Communication relay stations can keep their dishes and antennae constantly pointed in the
same direction. They do not need complicated tracking mechanisms to follow a satellite that is
moving relative to the Earth.
š
Communication facilities are not lost when the satellite moves out of the “line of sight”.
Geostationary satellites are also extremely useful for maintaining continuous surveillance on the
same part of the Earth’s surface for military reasons, to monitor weather patterns and for global
positioning reasons.
Currently there are more than 200 satellites in geostationary orbits above the Earth.
Geostationary Orbits
The Earth rotates from west to east with a period of 24 hours.
To maintain its position above a given point on the Earth’s surface the geostationary satellite must
also rotate in the same direction as the Earth (i.e. from west to east with a period of 24 hours).
Because the centre of the orbit must coincide with the centre of the Earth, and the satellite rotates
from west to east, geostationary satellites must be in an equatorial orbit (i.e. they rotate above the
equator of the Earth).
Thus, geostationary satellites can only be stationary above some point on the equator.
Notes on Geostationary Orbit
š
The period of a geostationary orbit is fixed at 24 hours.
Therefore, this determines the radius of the geostationary orbit, as period depends only on
radius as we showed in Section 3.2.1.
š
The height of a geostationary orbit is about 35,900km above the surface of the Earth.
We derive this result in Example 6 below. In
terms of general satellite orbits this is very high.
The vast majority of the satellites above the
Fig. 3.5
Earth are in orbits much lower than this.
Figure 3.5 portrays a geostationary satellite orbiting the Earth, drawn approximately to scale.
š
The height of a geostationary satellite above the Earth’s surface causes its own problems.
In the case of communication satellites.
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Signal strength must be high.
š
Receiving equipment must be quite sophisticated, in order to achieve a decipherable
signal from this distance.
š
There is a problem with time delays because of the distance that the signal has to travel.
Consider communication between Adelaide and London. The signal leaves Adelaide,
is beamed up to a satellite and then may be relayed via several other geostationary
satellites to London. The large distance travelled means that there is a noticeable time
delay before the signal reaches London. This is why, when watching a TV interview
with someone overseas, we detect a hesitation before the interviewee responds to a
question.
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66 CHAPTER
Key Ideas3
Stage 2 Physics
In the case of surveillance and weather satellites.
š
The height of the orbit limits the amount of close up detail that can be observed.
š
It is not possible to gain a good view of the surface of the Earth at high latitudes (i.e.
close to the poles).
Because the satellite is positioned above the equator, when viewing regions near the
equator from a geostationary satellite, we are viewing the surface of the Earth at an
angle of 90“ to the surface.
Because of the curvature of the Earth’s surface, when looking at high latitudes from a
geostationary satellite, the viewing angle is not at 90“ to the surface but gets closer and
closer to 0“ as we view nearer to one of the poles.
Thus, for these regions of the Earth we get a very unclear and distorted image.
Example 6
Calculate the height of a satellite that is in geostationary orbit above the Earth.
The mass of the Earth ME = 5º977—1024kg, the equatorial radius of the Earth rE = 6º378—106m.
Period T
24 hrs
24 — 3600 s
8 º 64 —10 4 s
v
v2
i.e.
4S 2 r 2
T2
r3
r
GM E
r
GM E
r
GM E
2Sr
as v
r
T
GM E T 2
4S 2
6 º 673 — 10 11 — 5 º 977 — 10 24 — 8 º 64 2 — 10 8
4S 2
3
75 º 4175 — 10 21
4 º 225 — 10 7 m
This is the radius of the satellite’s orbit.
The height of the orbit above the surface is given by
h
r rE
42 º 250 — 10 6 6 º 378 — 10 6
35 º 872 — 10 6 m
35,872 km
3.3.3 Direction of Launch of Equatorial Orbit Satellites
Consider a satellite in an equatorial orbit at a height of 1000km above the Earth’s surface.
It can be shown that the speed of this satellite in its orbit is approximately 7,350ms–1. To maintain
a stable orbit at this height the satellite must have this speed. This speed is fixed and it does not
depend on whether the Earth is stationary, or whether it is spinning on its polar axis below the
satellite.
The Earth is rotating on its polar axis with a period of 24 hours. It can be shown that a point on
the equator has a linear speed of 465ms–1 from West to East.
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Gravitation GRAVITATION
and Satellites AND SATELLITES 67
Thus if we wish to launch this satellite in an equatorial orbit it already has a speed of 465ms–1
from West to East before it leaves the launch pad. Therefore if we launch the satellite in a West to
East direction we only need to give it an additional speed of 6885ms–1 relative to the Earth.
If the satellite were to be launched in an East to West direction we would have to give it an
additional velocity of 7815ms–1.
Therefore, it is advantageous to launch low-altitude equatorial-orbit satellites in a West to East
direction, as about 930ms–1 smaller velocity change must be imparted to the satellite, with
consequential savings in fuel etc.
3.3.4 Satellites in Polar Orbits
Geostationary satellites have several disadvantages for surveillance and meteorological (weather
monitoring) applications.
Because of the great height of the orbit above the Earth’s surface the amount of fine detail that can
be observed is limited. Because these satellites are located above the equator geostationary
satellite images of polar regions are distorted because of the low angle at which the satellite sees
the region.
Low altitude polar orbit satellites are used to overcome many of these disadvantages. These
satellites rotate around the Earth in North-South orbits, passing over the Earth’s North and South
Poles, at altitudes of approximately 850km above the Earth’s surface. They orbit the Earth
approximately 14º2 times per day.
As the satellite in the polar orbit depicted in Figure 3.6
rotates about the Earth it surveys and photographs a northsouth strip of the Earth’s surface.
Meanwhile the Earth is rotating from West to East under
the satellite.
Therefore, when the satellite has completed one revolution
(about 1hr and 40mins later) it will now pass over a strip
approximately 2800km to the west of the previous one.
The photographic images of these strips can then be
pieced together to produce an image of a much larger area.
Rotation
of the Earth
The width of the field of view of these satellites on the
Fig. 3.6
Earth’s surface is about 2900km. Therefore successive
orbits give slightly overlapping images, thus making the job of piecing them together easier.
The United States has two fully functional polar orbiting weather satellites. They are called
NOAA 12 and NOAA 14. They orbit the Earth with periods of approximately 102 minutes at
altitudes between 820km and 860km. Thus, they complete about 14º2 orbits per day.
Their orbits are not truly polar. Instead of being inclined at 90“ to the equator they are inclined at
98“ to the equator. This means they are in retrograde orbits – they orbit in the opposite direction
to which the Earth turns.
The advantage of this particular orbit is that the satellites cover (i.e can “see”) the same part of the
Earth’s surface at the same time each day, and they do this day after day for the life of the satellite.
This kind of orbit is called a sun synchronous orbit.
Low altitude polar orbit satellites have the advantage that they are photographing areas of the
Earth virtually directly beneath them (therefore with minimal distortion) from a low altitude. This
enables analysts to see great detail. In addition one of these satellites will cover and observe every
point the Earth’s surface once each day.
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68 CHAPTER
Key Ideas3
Stage 2 Physics
3.4 Exercises
Conceptual Questions
1
The value of gravitational acceleration is the same
throughout the universe. Is this true or false?
Explain your answer.
2
The mass of an apple in the greengrocer’s shop is
0º2kg. With what force does the apple attract the
Earth?
3
An observer, stationed on the Moon, has a very highpower telescope and, with it, he observes an apple
fall off a tree down to the ground on Earth. If the
apple is attracting the Earth, why did he not see the
Earth fall towards the apple?
4
The orbit of the Earth about the Sun is not truly
circular. It is slightly elliptical. The closest distance
(at perihelion) is 1º496—1011m and the furthest
distance (at aphelion) is 1º521—1011m. Is the Earth
travelling faster at perihelion or at aphelion? Explain
your answer.
5
Explain why it takes more fuel for a spacecraft to
travel from the Earth to the Moon than for the return
journey.
6
It was proposed to put military satellites carrying
bombs in orbit around the Earth. By releasing it at
the correct time and location, one could drop a bomb
on any point on the Earth’s surface. Is this feasible?
7
The polar diameter of the Earth is less than the
equatorial diameter. Will the value of g be greater at
the North Pole or at the Equator? Explain your
answer.
8
At high altitude the value of g is less than at the
Earth’s surface. As a body falls towards the Earth’s
surface the value of g that it experiences increases.
Consider a body continuing to fall down a very deep
vertical mineshaft below the Earth’s surface. Does
the value of g continue to increase as the body
approaches the centre of the Earth? Explain your
answer.
9
My mass is 100kg. What is my weight on the
Earth’s surface? What would be my weight at the
centre of the Earth (if I could survive there)?
10 The force of gravity acting on a body on the Earth’s
surface is directly proportional to the mass of the
body. Therefore, explain why heavier bodies do not
fall faster than light ones.
11 A satellite is orbiting the Earth. A piece breaks off
an antenna on the satellite. Describe the subsequent
motion of this piece.
12 When a satellite is moving in a circular orbit there is
a gravitational force F acting on the satellite. In a
given time the satellite moves a certain distance s.
Therefore work is done on the satellite (as W = Fs).
Therefore the satellite must gain energy. Therefore
the satellite’s speed must increase (as kinetic energy
K= ½mv2). Thus it is impossible for a satellite to
move in a circular orbit with constant speed.
Find and explain the flaw in the above argument.
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13 “ACME Fine Foods” advertises that their prices are
the same at all their stores throughout the world.
Therefore explain why it is better to buy 1kg of
freshly sliced leg ham at their Darwin store than at
their South Pole store.
14 At night you are on the side of the Earth that is
facing away from the Sun, while during the day you
are on the side of the Earth that is facing the Sun. If
you had a very sensitive bathroom scale, would you
weigh more at midday or at midnight? Explain.
15 Consider four bodies of equal mass. Body A is a
large solid sphere, body B is a small solid sphere,
body C is a large hollow sphere and body D is a
small hollow sphere. A particle P is placed, in turn,
an equal distance from the centres of each of the
above spheres. Rank the four spheres in the order of
the magnitude of the gravitational force that they
would exert on P.
16 If the mass of the Earth were to suddenly double,
what effect would this have on the orbit of the
Moon?
17 Four students are explaining satellite motion.
(A) Ianthe says: “The satellite is no longer in the
Earth’s gravitational field and is no longer
attracted to the Earth. The net force on the
satellite is zero and it is kept in orbit by its
tangential velocity v”.
(B) Jim says: “The satellite is no longer in the
Earth’s gravitational field and the force of
gravity on it is negligible. There is, however, an
inward centripetal force mv2/r which maintains
the satellite in its orbit”.
(C) Kate says: “There is only the force of gravity
acting on the satellite and this causes an inward
acceleration of v2/r”.
(D) Leo says: “There is a gravitational force acting
on the satellite, but it is kept in its orbit by an
outward force mv2/r which cancels the effect of
the gravitational force”.
Which of the above explanations is correct?
18 A gravitational force is acting between two bodies.
What is its direction?
19 Is it possible to have a geostationary satellite rotating
from East to West.
20 Why is there not a satellite permanently stationed
above Adelaide?
21 A mistake was made in positioning a geostationary
satellite and it was placed in an orbit above the
equator that was 1km too high. Is the satellite
moving West to East relative to the Earth or East to
West?
Gravitation GRAVITATION
and Satellites AND SATELLITES 69
Descriptive / Explanatory Questions
1
Given the mass and the radius of some planet,
explain how you could calculate the value of g, the
gravitational acceleration, at the surface of the
planet.
2
Explain how the gravitational force between two
bodies is consistent with Newton’s third law.
3
Show that the speed v of a satellite orbiting a larger
body of mass M in a stable circular orbit of radius r
is given by v
GM
.
r
4
Prove that the period of a satellite in stable circular
orbit is independent of the mass of the satellite and
dependent only on the mass of the central body and
the radius of the orbit.
5
Explain why a geostationary satellite must move in
an orbit that coincides with the equatorial plane of
the Earth.
6
Is it possible for a satellite to move in a stable orbit
around the Earth, with the centre of the orbit not
coinciding with the centre of the Earth? Explain.
7
Explain why it is advantageous to launch satellites,
which are to circle the Earth in an equatorial orbit, in
a West to East direction in preference to an East to
West direction.
8
Explain why low-altitude polar orbits are preferred to
geostationary orbits for meteorological and spying
surveillance. In your answer include three disadvantages of geostationary orbits for these purposes
and three advantages of low-altitude polar orbits.
9
An astronaut of mass 90kg flies in a direct line from
the Earth to the Moon. On Earth g = 9º80ms–2 while
on the Moon g = 1º62ms–2. Describe and explain
how his weight changes during the flight.
14 A satellite is moving in a circular orbit of radius r
and period T. Show that the radius and period of the
4S2
r where g is the
orbit are related by T 2
g
gravitational acceleration at the satellite’s orbit
height.
15 At midnight on December 31, 2999 the Masters of
the Universe plan to decrease the universal
gravitational constant G from its current value of
6º67—10–11Nm2kg–2 to 4º0—10–11Nm2kg–2. Explain
what effect you expect this to have on each of the
following.
(1) The value of gravitational acceleration g on the
surface of the Earth.
(2) The radius of the Earth’s orbit about the Sun.
(3) The length of the year.
(4) The diameter of the Earth.
10 The planet Jupiter has 12 moons. Explain what
measurements an astronomer could take to determine
the mass of Jupiter.
11 For satellites moving in low polar orbits there are
still some residual traces of atmosphere at the height
of the satellite. Describe and explain the effects of
air resistance on the motion of the satellite.
12 Photographs of astronauts in the space shuttle or on
space station Mir show them in a state of weightlessness, i.e floating around the inside of the vehicle.
However at a height of 1000km above the surface of
the Earth the magnitude of gravitational acceleration
is g = 7º3ms–2. Explain how it is possible for them to
attain this weightless state at this altitude.
13 The diagram shows a
M
mass M and five smaller
equal masses m. The
smaller masses are
evenly spaced along the
line AB. What is the
A
B
direction of the net
m
m
m
m
m
gravitational force on M
due to the five smaller masses? Explain your answer.
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70 CHAPTER
Key Ideas3
Stage 2 Physics
Analytical / Computational Questions
1
(1) Determine the gravitational force between an
80kg man and a 60kg woman if they are
standing 10m apart.
(2) What will be the gravitational force if they
move towards each other so that the distance
between their centres of mass is reduced to 1m?
(3) What will be gravitational force between the
80kg man and a 30kg child, if they are 10m
apart?
(4) What will be gravitational force between the
80kg man and a 30kg child, if they are 5m
apart?
10 How high above the surface of the Earth must we go
before the value of gravitational acceleration is
reduced to 4º9ms–2?
2
Find the gravitational force of attraction between the
Earth and the Sun given that the mass of the Earth is
5º977—1024kg, the mass of the Sun is 1º991—1030kg
and their mean distance apart is 1º496—1011m.
3
Two bodies, of mass 5º0kg and 8º0kg respectively,
experience a gravitational force of 4º6—1010N.
Calculate the distance between their centres of mass.
12 The Russian space station MIR is in circular orbit
above the Earth at a height of 365º4km. Find
(1) its orbital speed;
(2) its period of revolution about the Earth;
(3) the number of times it circles the Earth in one
day.
4
The gravitational force of attraction between two
bodies, 3º0m apart, is 3º335—109N. The mass of one
body is double the mass of the other. Find the
masses of the two bodies.
5
The mass of the Sun is 1º991—1030kg and the mass of
Jupiter is 1º901—1027kg. The mean radius of
Jupiter’s orbit is 7º784—1011m. Find
(1) the gravitational force of attraction between
Jupiter and the Sun;
(2) Jupiter’s orbital speed;
(3) the length of a Jovian year (i.e. the time it takes
for Jupiter to complete one revolution). Give
your answer in Earth years.
6
The mass of Jupiter is 1º901—10 kg and its radius is
7º144—107m. Calculate the value of gravitational
acceleration, g, on the surface of Jupiter.
7
How far from the Earth must a spacecraft be along a
line towards the Moon so that the Moon’s
gravitational pull on the body balances that of the
Earth. The Moon is 3º844—108m away from the
Earth and the Earth’s mass is 81º33 times the mass of
the Moon.
8
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70
27
Fred’s planet has three moons named Alpha, Beta
and Gamma. Their relative masses are mE = 2mD and
mJ = 3mD. The relative radii of their orbits are given
by rE = 2rD and rJ = 3rD. The gravitational force of
attraction between Fred’s planet and Alpha is
2º50—1020N. Find
(1) the gravitational attraction between Fred’s
planet and Beta;
(2) the gravitational attraction between Fred’s
planet and Gamma.
On its journeys “where no man has gone before”, the
starship Enterprise discovers a new planet. Its mass
is double the mass of the Earth and its radius is ¾
that of the Earth. Determine the value of
gravitational acceleration on the surface of this
planet.
Essentials Text Book
11 A satellite is in circular orbit above the Earth. Its
orbital radius is 6º77—106m. Find
(1) its orbital speed;
(2) its period of revolution about the Earth;
(3) the number of times it circles the Earth in one
day;
(4) the value of gravitational acceleration, g, at this
satellite’s orbit.
13 A satellite, in circular orbit above the Earth, circles
the Earth 10 times in one day. Find
(1) its period;
(2) the radius of its orbit;
(3) its orbital speed;
(4) its height above the Earth.
14 A satellite is in geostationary orbit above Earth.
(1) Explain what this means and state the period of
the satellite.
(2) Calculate the height of the satellite above the
surface of the Earth.
(3) Calculate the orbital speed of the satellite.
15 A satellite is in circular orbit about the Earth. It has
an orbital speed of 5595ms–1. Find
(1) its height above the Earth’s surface;
(2) its period of rotation.
16 The following table gives the value of gravitational
acceleration g at various distances r from the centre
of the Earth.
r (—106)m
g (ms–2)
(1)
(2)
(3)
(4)
(5)
6º4
9º80
7º0
8º19
8º0
6º27
9º0
4º96
10
4º01
1
Draw a graph of g against 2 .
r
What does your graph tell about the
proportionality relationship between g and r?
Use your graph to find the value of g at a
distance of 8º5—106m from the centre of the
Earth.
Use your graph to find the distance from the
centre of the Earth at which the value g is
3º0ms–2.
Use your graph to find the mass of the Earth.
The mass of Venus is about 0º8 times the mass
of the Earth and its radius is slightly less than
that of the Earth. On your graph draw a line
which would approximately represent similar
data for the planet Venus.
Gravitation GRAVITATION
and Satellites AND SATELLITES 71
Analytical / Computational Questions
17 The following table gives the orbital speeds v and the
altitudes h of satellites in circular orbits about the
Earth. Take the radius of the Earth to be 6º4—106m.
h (km)
500
1000
1500
2000
2500
v (—10 ms ) 7º60
7º34
7º11
6º89
6º69
3
–1
(1) Find the orbital radius, r, of each satellite and
1
draw a graph of v against
(2)
(3)
(4)
(5)
r
.
What does your graph tell you about the
proportionality relationship between v and r?
Use your graph to find the orbital speed of a
satellite that is 3000km above the surface of the
Earth.
Use your graph to find the orbital radius of a
satellite with an orbital speed of 5000ms–1.
Use your graph to find the mass of the Earth.
The mass of Fred’s planet (see question 8) is
approximately four times the mass of the Earth
and its radius is the same as the radius of the
Earth. On your graph draw a line which would
represent similar data for Fred’s planet.
18 At the beginning of the seventeenth century, more
than half a century before Isaac Newton published
his laws of motion and the law of universal
gravitation, a German astronomer Johannes Kepler,
after examining and analysing detailed astronomical
data collected by Tycho Brahe, published three laws
of planetary motion. Kepler’s third law states that
the square of the period of a planet’s motion about
the Sun is directly proportional to the cube of the
radius of its orbit.
Newton could show that this law, and the other two
laws, could be derived mathematically from the law
of universal gravitation and the laws of motion, and
he used this fact as evidence in favour of his law of
universal gravitation.
(1) Prove Kepler’s third law mathematically; i.e.
⎛ GM S ⎞ 2
show that r 3 ⎜
⎟T , where r is the radius
2
⎝ 4S ⎠
of a planet’s orbit, T is its period and MS is the
mass of the Sun.
The table below gives some planetary data for our
solar system. It gives the mean distance from the sun
r of five planets and the length of their orbital period
T about the sun (i.e. the length of their year) in Earth
years.
Planet
Mercury
9
r (—10 m)
57º9
Venus
Earth
Mars
108º2
149º6
227º9
Jupiter
778º3
T (years)
0º241
0 615
1º0
1º88
11º86
(2) Using the data for the five planets listed, draw a
18 (3) Use your graph to find the value of
r3
for the
T2
planets of our solar system.
(4) Use this ratio to determine the mass of the Sun.
19 Once a star has burnt out it can collapse under its
own gravitational force to become a neutron star or
“black hole” (so called because its gravitational pull
is so strong that even light cannot escape from it).
Consider our Sun collapsing into a neutron star of
diameter 20km. Assuming that the mass of the Sun
at that time is 1º0—1030kg
(1) Find the values of gravitational acceleration g at
distances of 10m and 12m above the surface of
this neutron star.
(2) Deduce what would happen to your body if you
were to “fall” head first into a black hole.
20 An object on the Earth’s equator is subject to three
centripetal accelerations.
One towards the centre of the Earth due to the
rotation of the Earth about its axis.
One toward the Sun due to the rotation of the Earth
about the Sun.
One toward the centre of our galaxy due to the
rotation of the Sun about the centre of our galaxy.
Determine each of these centripetal accelerations as
multiples of g = 9º8ms–2.
[Use astronomical data from Appendix C plus the
following: Period of Sun’s rotation about the
galactic centre = 2º5—108yr, distance of Sun from
galactic centre = 2º2—1020m.]
21 A mass M is split into two parts, one of mass m and
the other of mass M – m. The two parts are then
separated by a distance d. Show that the gravitation
force between them is maximum if the two parts are
of equal mass.
[You need to know how to find the maximum value of
a quadratic function of the form y = –ax2 + bx to
solve this problem.]
22 Assume that all the stars in our galaxy are uniformly
distributed in a sphere about the galactic centre.
Also assume that the average mass of the stars is
equal to the mass of our Sun. Our solar system is on
the outer edge of or galaxy and the Sun rotates about
the galactic centre with a period and radius as given
in Question 20 above. Estimate the number of stars
in our galaxy.
23 An astronaut flies in a direct line from the Earth to
the Moon. Using the astronomical data in Appendix
C, determine his distance from the Moon when he is
truly weightless.
24 A cannonball is fired horizontally from a height of
2m above the surface of the Earth. Determine the
speed with which it must be projected so that it
travels around the Earth. Assume that the Earth is a
perfect sphere and neglect the effect of air resistance.
2
graph of r against T 3 to show this
proportionality.
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