Download - Maths and Astronomy Comenius Why Maths
Maths and Astronomy
Maths and Astronomy
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THIS EBOOK WAS PREPARED
AS A PART OF THE COMENIUS PROJECT
WWHHYY MMAATTHHSS??
by the students and the teachers from:
BERKENBOOM HUMANIORA BOVENBOUW, IN SINT-NIKLAAS ( BELGIUM)
EUREKA SECONDARY SCHOOL IN KELLS (IRELAND)
LICEO CLASSICO STATALE CRISTOFORO COLOMBO IN GENOA (ITALY)
GIMNAZJUM IM. ANNY WAZÓWNY IN GOLUB-DOBRZYŃ (POLAND)
ESCOLA SECUNDARIA COM 3.º CICLO D. MANUEL I IN BEJA (PORTUGAL)
IES ÁLVAREZ CUBERO IN PRIEGO DE CÓRDOBA (SPAIN)
This project has been funded with support from the European Commission.
This publication reflects the views only of the author, and the
Commission cannot be held responsible for any use which may be made of the
information contained therein.
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Astronomy is a science that studies celestial objects, such as moons, planets and
stars. It involves other sciences, like physics, chemistry and mathematics, in order to
explore the history and evolution of these celestial objects. It is one of the oldest
sciences, but one of the most important ones nowadays. The connection between
Maths and Astronomy is very important, in order to calculate the volume, density,
distance to Earth, to know the orbits and other useful information about the objects
in the universe. ! Astronomy is a fascinating science, from the distances to and inter-
workings of stars and planets.
Most of mathematical skills are
grade school level arithmetic
skills, so it is not needed to be a
mathematics major to
understand the mathematical
concepts necessary to do well in
an introductory astronomy
class. One of the first
mathematical challenges we
find ourselves facing in
astronomy is dealing with very
large numbers, Basic arithmetic rules of manipulation for addition, subtraction,
multiplication, and division apply to numbers written in scientific notation.
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Jupiter or the Sun?
Jupiter is the biggest planet on the Solar System and the Sun is the
biggest star. But which one of them is bigger? To do that, I will
calculate the volume of the two celestial objects. Although Jupiter
and the Sun are not perfect spheres, I will calculate the volume of
them as if they were. To calculate the volume of a sphere, we use the
mathematical expression:
where R represents radius. So, we need the radius of Jupiter and Saturn.
Jupiter radius: 71,492 km
Sun radius: 696,342 k
Jupiter volume:
Sun volume:
How many times is the Sun bigger than Jupiter?
To do that, we just need to do a simple division equation - we
divide the Sun volume (the bigger one) by the Jupiter volume
The sun is approximately 928 times bigger than Jupiter.
How many “Europes” fit in the Sun?
To calculate how many Europes fit in the Sun, we need to
calculate the area of the Sun and of Europe. The area of Europe is 10180000, and we
calculate the area of the Sun. To calculate the area of a sphere, we use the
mathematical expression:
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Now, we just divide the area of the Sun by the area of
Europe.!
Almost 60000 “Europes” fit in the Sun.
How many days it would take if I wanted to go to the Moon by plane?
Imagine is planes could fly on space - how many days it would take to get to the
Moon? To calculate that, we need the distance between the Earth and the Moon and
the average speed of the plane.
Distance between Earth and Moon: approximately 400000 km
Average speed of a Boeing 747: 920km/h
In order to calculate the days, we need to convert the hours to days:
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Here we compare
all the numbers and
the statistics about
the characteristics
of the planets, the
Moon and the Sun
but as The sun has
big numbers,
sometimes it
doesn't appear in
the graphs.
RADIUS
Looking at the radius we know which is the
biggest, and as we can see it's Jupiter. The
planets which are close to the Sun, are
smaller than the planets which are far from
the Sun.
MASS
Now it´s time to compare the mass of
all the planets. Jupiter has the biggest
mass. The mass of Jupiter is twice that
of all the other planets combined.
Mercury has a mass of 3.30·1023 kg,
making it the lightest planet in our
Solar System.
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AGE
All solar system was formed at
the same time so all the
planets have the same age;
four and a half million years.
SURFACE
The data about the radius is similar to the
surface's data because to calculate the surface
we need the radius. The surface shows us
which is the biggest and the smallest planet.
And as we can see Jupiter is the largest one
and Mercury the smallest one.
DISTANCE TO THE SUN
As we know, Uranus and Neptune
are very far from the Sun. Here we
have the distance to the Sun in
kilometers.
DISTANCE OF ALL PLANETS FROM EARTH
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MOON’S NUMBER
In this graph we can see how many moons
different planets have, for example
Mercury and Venus have no moons. The
Earth has only one moon. Uranus,
Neptune and Saturn have a lot of moons
and Jupiter is the planets with more
moon's number with 67 moons.
TRANSLATION MOVEMENT
Neptune has a longest translation movement, his duration is 60000 Earth´s days.
ROTATION MOVEMENT
A day is the length of time it takes
for a planet to complete one
rotation on its axis – 360°. Since all
of the planets rotate at different
speeds, the length of a day on each
one differs. Compared to Earth,
Mercury has a very long day. A day
on Mercury takes 58 days and 15
hours in Earth days. Venus is the
slowest moving planet. Venus has
the longest day of any planet in our
Solar System. It is about 225 Earth
days for the planet to orbit the Sun.
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DENSITY
Density (ρ symbol) is a scalar relating to
the quantity of mass in a given volume of
a substance. The average density is the
ratio of the mass of a body to its volume.
Just like the size of the planets, the
density of the planets varies widely. All of
the 4 inner planets – the planets closest
to the Sun – are much denser than the
four outer planets.
GRAVITY
Gravity is an attracting force that is
present in all objects in the universe. It
differs depending on what planet we are
on. This is because the planets vary in size
and mass. In our solar system, the planet
with the greatest gravity is Jupiter. and
lowest gravity planet is Mercury.
ESCAPE VELOCITY
Escape velocity is the speed that an
object needs to be traveling to break
free of a planet or moon's gravity
well. The escape velocity is
determined by the gravity of the
planet which in turn is determined
by the mass and size of the planet.
The escape velocity from the moon
is so much smaller than from the
earth that it's no wonder that the
moon can't keep an atmosphere.
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MAX. AND MIN TEMPERATURE
Many people know that most of the planets
in our Solar System have extreme
temperatures unsuitable for supporting
life. Mercury is the planet closest to the
Sun, so one would assume that it is a
burning furnace. While the temperature on
Mercury can reach 465°C, it can also drop
to frigid temperatures of -184°C. Venus,
the second closest plant to the Sun, has the
highest average temperatures of any planet
in our Solar System, regularly reaching
temperatures over 460°C. Venus is so hot
because of its proximity to the Sun and its
thick atmosphere. With temperatures
dropping to -218°C in Neptune’s upper
atmosphere, the planet is one of the
coldest in our Solar System.
ELEMENT’S NUMBER
The numbers of elements that form the different planets. The Moon has 25 elements
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The sundial is the oldest known device for the measurement of time. It uses the
motion of the apparent Sun to cause a shadow or a spot of light to fall on a reference
scale indicating the passage of time.
Every sundial is composed of various
parts, indeed we can find:
- a style that casts the shadow (or the
light) of the Sun; it may be a thin rod or a
sharp, straight edge
- a gnomon, that is the terminal point
of the style. Sometimes it is just a hole in a
wall
- some lines drawn on the dial plate:
hour lines permit us to tell the time from
the shadow cast by the style; declination
lines permit us to determine the date from
the shadow cast by the style.
It is impossible to determine exactly when the first sundials appeared and who were
their inventors.
The Roman author, architect and engineer Vitruvius wrote about "ἡ γνωμονικὴ
τέχνη" in his famous work entitled De Architectura (1st century BC).
And "ἡ γνωμονικὴ τέχνη" means "the art of making sundials", in other words what we
call gnomonics, which is the science of sundials.
In the ninth book of De
Architectura Vitruvius listed all
thirteen known types of dials,
together with their inventors:
Aristarcus of Samos and Eudoxus of
Cnidus are just a couple of names that
we can mention.
Moreover Vitruvius credited the
Babylonian astronometer Berossus,
lived in the third century BC, with the
invention of the sundials: his
hemicyclium had a truncated,
concave, hemispherical surface.
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But the earliest prototypes of sundials were Babylonian, Egyptian and Greek.
Sure enough Babylonians and Egyptians built obelisks whose moving shadows
formed a kind of sundial, enabling citizens to divide the day into two parts by
indicating noon.
The Egyptian red
granite Obelisk of
Montecitorio was
brought to Rome in
10 BC to be used as
the gnomon of the
Solarium Augusti in
Campus Martius: it
cast its shadow on a
marble pavement
inlaid with a gilded bronze network of lines, by which it
was possible to read the time of the day according to the
season of the year.
We can observe several different
types of sundials.
In the horizontal sundial (also
called a garden sundial), the plane
that receives the shadow is
horizontally aligned. The line of
shadow does not rotate uniformly on
the dial face; so the hour lines are
spaced according to the rule:
tanθ = sinλ tan(15°×t), in which: θ is
the angle between a give hour-line
and the noon hour-line;
λ is the geographical latitude and t is
the number of hours before or after
noon.
In the equatorial dial the planar surface
that receives the shadow is exactly
perpendicular to the gnomon's style and it
is parallel to the equator of the Earth and
of the celestial sphere, so the Sun's
apparent rotation about the Earth casts an
uniformly rotating line of shadow from the
gnomon on the equatorial plane.
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The Sun rotates 360° in 24 hours, so the hour-lines on an equatorial dial are all
spaced 15° apart (360/24).
In the vertical dial, the shadow-receiving plane
is aligned vertically; as usual, the gnomon's
style is aligned with the Earth's axis of
rotation.
As in the horizontal dial, the line of shadow
does not move uniformly on the face.
If the face of the dial points directly south, the
hour lines are spaced according to the rule
tanθ = cosλ tan(15°×t).
Motion of the Earth
The Earth travels around the Sun in an ellipse,
and its speed is fastest when it's closest to the
Sun, in January, and slowest when it's farthest
away from the Sun, in July, according to
Kepler's second law of planetary motion. At
the same time, it rotates from the west
towards the east around its own axis.
An object's axial tilt is the angle between its
equatorial plane and orbital plane. The Earth has an
axial tilt of about 23.45°, known as the obliquity of
the ecliptic. When a hemisphere is tilted toward the
Sun, it has longer days and shorter nights.
Apparent motion of the Sun
An excellent approximation assumes that the Sun revolves around a stationary Earth
on the celestial sphere, which rotates every 24 hours about its celestial axis. The Sun
changes its position on the celestial sphere, being at a positive declination in spring
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and summer, and at a negative declination in autumn and winter, and having exactly
zero declination (being on the celestial equator) at the equinoxes.
Equation of time
The equation of time describes the
discrepancy between apparent
solar time and mean solar time.
Indeed apparent time can be
ahead (fast) by as much as 16 min
33 s (around 3rd November), or
behind (slow) by as much as 14
min 6 s (around 12th February).
This depends on the obliquity of
the ecliptic and the eccentricity of
the Earth's orbit around the Sun.
The graph of the equation of time
is obtained by summing the sine
curve of the obliquity to the sine curve of the eccentricity.
But the equation of time can be represented
graphically, too: it's the east or west component of
the analemma, an eight-shaped figure, similar to
the lemniscate of
Bernoulli, that can
be placed on the
sundials to read
the mean time.
It is quite difficult
to imagine how
the analemma
could be, but you
can obtain it: you need a camera and a lot of
patience! Indeed if you take a picture of the Sun at
the same time each day, from the same place, you
can see that the shape traced out by the Sun over the
course of a year is an analemma. In the summer the
Sun appears at its highest point in the sky, and
highest point in the analemma. Instead in the
winter, the Sun is at its lowest point.
But what is the lemniscate of Bernoulli? It is a curve «shaped like a figure 8, or a
knot, or the bow of a ribbon» (as Jacob Bernoulli wrote in Acta Eruditorum), and
defined from two given points F1 and F2 (that we call foci), at distance 2a from each
other as the locus of points M so that MF1·MF2 = a².
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How to make a vertical sundial
What do we need?
- a wall facing South, where we
can place the dial;
- a thin rod, to be used as the
style;
- a compass,
- a square ruler
The style.
We have to find the value of the angle between
the style and the dial plate.
The gnomon's style must be parallel to the
Earth's axis of rotation, so the direction of the
style crosses perpendicularly the Equator,
because it crosses the axis of the Earth. If we
look at the triangle that has got the style, the
centre of the Earth and the intersection
between the Equator and the direction of the
style as vertices, we see that we can find the
value of the angle by subtracting the latitude of
the place from 90°.
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Now we can draw the noon line, that
is the vertical line traced from the
insertion of the sundial in the wall.
Then, we have to draw the other hour
lines, using the rule tanθ = cosλ
tan(15°×t).
Our sundial is ready to be used!
But we must remember that it tells the solar
time: so there are problems due to the
equation of time and the daylight saving
time during the lighter months!
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Determining the distance to objects in astronomy is one of the most important tasks
and yet it is one of the most difficult ones.
In the solar system, even the kilometer is inconveniently small to use as a unit. One
way scientists measure the distance between
planets is to use the distance from the Earth
to the Sun as the standard unit of
measurement. This distance is called an
Astronomical Unit, or 1 A.U. It is equal to
approximately 150,000,000 kilometers.
Incredibly, stars and galaxies are much
farther that even AU's become unwieldy and
light-years become the standard.
A lightyear (ly) is the distance light travels
in one year at its speed of 300,000 km/sec. If
we multiply this speed by the number of
seconds in a year, we get the distance:
A light-year is the distance light travels in
one year – 9.5 trillion km – and that light
travels 300,000 km per second.
Our galaxy, the Milky Way, is about
150,000 light-years across, and the
nearest large galaxy, Andromeda, is 2.3
million light-years away.
Some other distances in light years:
Object Distance in light years
Nearest Star (Proxima Centuri) 4.2
Sirius the dog star (the brightest star in the
sky) 8.6
centre of the galaxy approximately 30 000
Image from: www.mathscareers.org.uk
Image from:
Image from:
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If the angular diameter is not very large (which is
typically the case), problems of this type may be
done without the use of trigonometry. For small
enough α, tan α ≈ α when the angle is measured
in a common unit known as radians.
A parsec is defined as the distance at which the parallax angle is 1 arc second. A
parsec is approximately 3.26 light-years.
It is also common to define large multiples of the parsec,
1 kiloparsec (kpc) D 103 pc
1 megaparsec (Mpc) D 106 pc
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for very large distances. The following table summarizes some common astronomy
distance units and gives their size in meters.
Quantity Abbreviation Distance (km)
Astronomical unit AU 1.50·108
Light year ly 9.46·1012
Parsec PC 3.08·1013
Kiloparsec kpc 3.08·1016
Megaparsec Mpc 3.08·1019
We should remember that when multiplying numbers in scientific notation, we have
to multiply the number part, times ten to the power of the sum of the exponents.
For example:
When dividing numbers in scientific notation, we have to divide the number part. The
answer is multiplied by 10 to the power which is the difference between the
exponents.
For example:
Examples
1. Convert each number of light years to kilometers.
a) 6 light years
b) light years
c) light years
2. Neptune is 4500 million kilometers from the Sun. How far is this in AUs?
30 AU
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Neptune is 30AU far from the Sun.
3. The second brightest star in the sky (after Sirius) is Canopus. This yeallow-white
supergiant is about 1.141016 kilometers away. How far away is it in light years?
Answer: It’s about light years.
4. Regulus (one of the stars in the constellation Leo the Lion) is about 350 times
brighter than the Sun. It is 85 light years away from the Earth. How far is this in
kilometers?
Answer: Regulus is away from the Earth.
5. Calculate the diameter of the Milky Way galaxy in kilometers assuming that its
radius is 50 ly.
Diameter of the Milky Way galaxy (D):
D = 2 · 50 · 9.46 · 1012 = 9.46 · 1014 km
6. The distance from earth to Pluto is about 28.61 AU from the earth. How many
kilometers is it from Pluto to the Earth?
Answer: Pluto is about away from the Earth.
7. A star is 4.3 light years from Earth. How many parsecs is this?
Answer: It’s parsecs.
8. Our Galaxy is 100,000 ly wide. How many meters
wide is it?
Answer: It’s meters.
Image from: www.hplusmagazine.com
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TT OO GG RR AA VV II TT YY AA NN DD SS UU RR FF AA CC EE GG RR AA VV II TT YY ??
Gravitation is a natural phenomenon by which all physical bodies attract
each other. It is most commonly experienced as the agent that gives weight to objects
with mass and causes them to fall to the ground when dropped.
Gravity is the force with the Earth, Moon, or other massive body attracts an
object towards itself. By definition, this is the weight of the object. All objects on
Earth experience a force of gravity, which is, directed “downward” towards the center
of the object the Earth.
In 1687, English mathematician Sir Isaac Newton
published “Principia”, which hypothesizes the inverse-square
law of universal gravitation.
He wrote: “I deduced that the forces which keep
the planets in their orbs must [be] reciprocally as
the squares of their distances from the centers
about which they revolve: and thereby compared
the force requisite to keep the Moon in her Orb with
the force of gravity at the surface of the Earth; and
found them answer pretty nearly”.
Newton's law of universal gravitation:
Where G is a constant equal to
m = mass of the body 1,
M = mass of body 2,
r = radius between the two bodies.
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The force due to gravity is given by
Equating both the force formula we get
Hence acceleration due to gravity formula is given as
It is used to find the acceleration due to gravity anywhere in space. On earth the
acceleration due to gravity is 9.8
.
Examples
a) Calculate the acceleration due to gravity on Earth.
Given:
r =
M =
so
Check the units:
a) Calculate the acceleration due to gravity on the Moon. The Moon’s radius is
and its mass is
Remember,
so
Gravity differs depending on what planet you are on. This is because the planets vary
in size and mass.
b) Calculate the acceleration due to gravity on Mercury. The radius of Mercury is
about 2.43106 m and its mass is 3.181023 kg.
Given:
6
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Answer: The acceleration due to gravity on Mercury is
c) Calculate the acceleration due to gravity on Venus. The radius of Venus is
about 6.06 106 m and its mass is 4.88 1024 kg.
Given:
Answer: The acceleration due to gravity on Venus is
Planet Radius (m) Mass (kg) g
Mercury 3.6
Venus 8.9
Earth
Mars
Jupiter 26.0
Saturn 11.2
Uranus 3.61
Neptune 13.3
Examples
1. A 6.2 kg rock dropped near the surface of Mercury reaches a speed of
in 5.0s.
a) What is the acceleration due to gravity near the surface of Mercury?
b) Mars has an average radius of 2.43 m. What is the mass of Mercury?
a)
The acceleration due to gravity is 3.61
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b)
The mass of Mercury is
2. In The Little Prince, the Prince visits a small
asteroid called B612. If asteroid B612 has a radius of
only 20.0 m and a mass of 1.00 104 kg, what is the
acceleration due to gravity on asteroid B612?
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VV II II .. WW EE II GG HH TT OO NN OO TT HH EE RR PP LL AA NN EE TT SS
Some planets have a stronger gravity than Earth's, some have weaker. On a planet
with a weaker gravity, we would be able to carry more mass and jump higher. When
Astronauts visited the Moon, which has one sixth of our gravity, they bounced around
on the surface as if they were floating with each step. On a planet with a stronger
gravity, we might be forced to our knees by just our own weight.
Weight can be defined by the gravitational force on an object, with a known mass.
This means that in order to obtain the weight of an object we take the mass of an
object, and multiply that by the acceleration of gravity. The mass of an object is the
same no matter where it exists, while the weight of an object changes depending on
what the gravitational field strength where the object exists. Weight is measured in
Newtons (N).
Weight is a force caused by the pull of gravity acting on a mass:
1. The Earth's moon is the only heavenly body
that people have walked on. The gravity of the
moon is 17% of Earth's gravity. To calculate my
weight on the Moon, I multiplied my weight by
0.17.
Answer: My weight on the Moon is about 98N.
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2. Mercury is the smallest planet, and the planet closest to the sun. The gravity of
Mercury is 38% of Earth's gravity. To calculate my weight on Mercury, I
multiplied my weight by 0.38.
Answer: My weight on Mercury is about 220 N.
3. Venus is known as the “Cloudy Planet” because it is covered with thick, yellow
clouds. The gravity of Venus is 90% of Earth's gravity. To calculate my weight
on Venus, I multiplied my weight by 0.9.
Answer: My weight on Venus is about 520N.
4. Mars is known as the “Red Planet” because the soil is filled with orange-red
particles. The gravity of Mars is 38% of Earth's gravity. To calculate my weight
on Mars, I multiplied my weight by 0.38.
Answer: My weight on Mars is about 220 N.
5. Jupiter has more moons than any other planet. So far, scientists have
discovered 63! The gravity of Jupiter is 234% of Earth's gravity. To calculate
my weight on Jupiter, I multiplied my weight by 2.34.
Answer: My weight on Jupiter is about 1353 N.
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6. Saturn is known as the “Ringed Planet” because it has colourful rings made of
rock and ice. The gravity of Saturn is 108% of Earth's gravity. To calculate my
weight on Saturn, I multiplied my weight by 1.08.
Answer: My weight on Neptune is about 624 N.
7. Neptune is a blue planet with extremely strong winds. The gravity of Neptune
is 112% of Earth's gravity. To calculate my weight on Neptune, I multiplied my
weight by 1.12.
Answer: My weight on Neptune is about 648 N.
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Planet/star Mass(kg) Gravity ( relative
to Earth) Weight (N)
Moon 59 17%
Mercury 59 38%
Venus 59 90%
Mars 59 38%
Jupiter 59 234%
Saturn 59 108%
Neptune 59 112%
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The gravitational field strength, g, of a planet is the weight per unit mass of an object
on that planet. It has the units N/kg e.g.
Earth g = 9.8 N/kg
Mars g = 3.69 N/kg
Moon g = 1.6 N/kg
The weight of an object can be calculated on different planets,if we know that object's
mass and the gravitational field strength of the planet. We can also calculate weight
using the following formula.
Where F weight and it is measured in Newtons (N)
m mass and it is measured in kilograms (kg)
g gravitational field strength, g, of a planet is the weight per unit mass of an object
on that planet. It has the units N/kg.
Planet/star Mass(kg) Gravitation (m/s2) Weight (N)
Earth 59 9.8 578.2
Moon 59 1.6 94.4
Mercury 59 3.73 220.07
Venus 59 8.87 523.33
Mars 59 3.69 217.71
Jupiter 59 25.9 1528.1
Saturn 59 11.19 660.21
Uranus 59 8.69 512.71
Neptune 59 11.28 665.52
Examples of some calculations:
For Earth:
For Mercury:
For Jupiter:
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For Neptune:
1. How much more would you weigh on Jupiter than Earth?
For Jupiter we have:
For Earth we have:
Answer: I would weigh 949.9 N more on Jupiter than Earth.
2. How much less would you weigh on Pluto than Earth?
For Earth we have:
For Pluto we have:
Answer: I would weigh 531 N less on Pluto than Earth.
3. Would you weigh more on the Earth's moon, or on Mercury?
For Earth's moon we have:
For Mercury we have:
Answer: I would weigh more on Mercury.
4. Somewhere you place a 7.5 kg pumpkin on a spring scale.
If the scale reads 78.4 N, what is the acceleration due to
gravity at that location?
Given: Calculate:
Answer: I think it is on the Earth.
Here you can watch the film prepared by the Polish school: LINK
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V I I I . H O W F A S T D O T H E P L A N E T S M O V E ?
On the average highway, a car travels at
approximately 100 km/hour. How fast
do the planets travel as they revolve
around the Sun?
1. If the Earth day comprises 24 hours we calculate the number of Earth hours
required to complete the orbit of each planet.
2. We know the circumference of each planet’s orbit and the number of hours to
complete that orbit. We use this information to calculate the planet’s speed.
1. For Mercury we have:
t = 88·24 = 2112
The velocity is:
v - velocity
s - circumference of an orbit
t - time to complete orbit
2. For Venus we have:
t = 225 · 24 = 5400
The velocity is:
3. For Earth we have:
t = 365 · 24 = 8760
The velocity is:
4. For Mars we have:
t = 687 · 24 = 16488
The velocity is:
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5. For Jupiter we have:
t = 4333 · 24 = 103992
The velocity is:
6. For Saturn we have:
t = 10759 · 24 = 258216
The speed is:
7. For Uranus we have:
t = 30685 · 24 = 736440
The velocity is:
8. For Neptune we have:
t = 60189 · 24 = 1444536
The velocity is:
The following table gives information about the orbits of the eight planets of the solar
system. The force of gravity makes the planets move in orbits that are nearly circular
around the Sun.
Planet Circumference
of orbit (km)
Earth time
to complete
orbit (days)
Earth time
to complete
orbit (hours)
Planet
Speed
(km/h)
Mercury 5.79·107 88 2112 2.7·10⁴
Venus 1.08·108 225 5400 2·10⁴
Earth 1.50·108 365 8760 1.7·10⁴
Mars 2.28·108 687 16488 1.38·10⁴
Jupiter 7.78·108 4333 103992 7.48·103
Saturn 1.43·109 10759 258216 5.5·103
Uranus 2.87·109 30685 736440 3.89·103
Neptune 4.50·109 60189 1444536 3.1·103
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Questions:
1. Which planet travels fastest?
Answer: The fastest planet is Mercury.
2. Which planet travels slowest?
Answer: The slowest planet is Neptune.
The closer a planet is to the Sun, the faster it travels in its orbit and the less time it
takes to complete a full trip around the Sun.
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II XX .. HH OO WW OO LL DD WW OO UU LL DD YY OO UU BB EE OO NN AA NN OO TT HH EE RR
PP LL AA NN EE TT ??
Have you ever wondered how old you would be if you lived on another planet? Using
a simple calculation against the planetary year, we can do just that!
We should remember:
Day: the time taken for the Earth to rotate once about its axis.
Year: the time taken for the Earth to revolve once around the Sun
All the planets in the solar system revolve around the Sun in the same direction. On a
given planet, the “year” is the period of time this planet takes to complete one orbit
around the Sun.
If a year is described as the amount of time it takes for a planet to revolve around the
Sun, for the Earth it’s 365.25 days, then our age would be different on each planet.
A planetary year is the length of time it takes that planet to revolve around the Sun.
The planets revolve around the sun in different amounts of time, so a "year" on each
planet is a different amount of time. The farther a planet is from the sun, the longer
its year.
Using the chart below I can figure out how old I would be.
Planet
Number of
days in a
planetary
year
- revolution
Multiply
your age by Years Days
Mercury 87.97 4.152 61 91
Venus 224.7 1.626 24 22
Earth 365.25 1 14 5401
Mars 687 0.53 7 5251
Jupiter 4.333 0.084 1 13032
Saturn 10.759 0.034 0.5 12312
Uranus 30.685 0.012 - 7499
Neptune 60.188 0.006 - 8025
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To work out your age on other planets, first you need to calculate how many Earth
days you have been alive (being careful of leap years). You can then calculate how
many days or years that would be on another planet by finding out how long that
planet takes to spin or orbit.
I was born on 3rd June 1999. Today is: 17 March 2014.
I am 5,401 days old (in days).
Mercury:
Calculations for Saturn:
Calculations for Venus:
For example, if I am 14 years ( 5401 days )
old here on Earth and I want to know my
age on Venus I have to divide the number
of days on the Earth by the number of
days in a planetary year of Venus:
Calculations for Mars:
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If we could live on another planet, our birthdays would take place more or less often
depending on the planet’s revolution period (the time taken to complete one full trip
around the Sun). On a few planets, we couldn’t even celebrate our first birthday
because we wouldn’t live long enough to give these planets time to complete one full
trip around the Sun!
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XX .. EE SS CC AA PP EE VV EE LL OO CC II TT YY
If you throw a rock straight up in the air, eventually it will come straight back down.
If you fire a gun straight up in the air, the bullet will travel higher than the rock but
will also eventually come straight back down.
Escape velocity is defined as a minimum velocity with which a body should be
projected so that it overcomes the gravitational pull of the earth.
What speed is required to escape the pull of Earth’s gravity?
In physics, escape velocity is the speed at which the kinetic energy plus the
gravitational potential energy of an object is zero. It is the speed needed to "break
free" from the gravitational attraction of a massive body, without further propulsion.
If the kinetic energy of an object launched from the Earth were equal in magnitude to
the potential energy, then in the absence of friction resistance it could escape from
the Earth.
This can be written mathematically as:
Re-arranging the equation to find v will give the escape velocity for the Earth.
The escape velocity (vesc) of a body depends on the mass (M) and the radius (r)
of the given body. The formula which relates these quantities is:
Escape velocity formula is helpful in finding escape velocity of any body or planet, if
mass and radius is known. It has wide applications in space calculations.
If a rocket is launched with the velocity less than the escape velocity, it will eventually
return to Earth.
If the rocket achieved a speed higher than the escape velocity, it will leave the Earth,
and will not return.
1. Determine the mass and radius of the planet you are on.
For Earth, assuming that we are at sea level, the radius is 6.38106 meters and the
mass is 5.971024 kilograms.
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We need the gravitational constant (G), which is 6.6710-11
. It is required to use
metric units for this equation:
Given:
Radius ( r ) = 6.38106 meters
Mass ( M )= 5.971024 kilograms
Gravitation constant = 6.6710-11
2. Using the above data, calculate the required velocity needed to
exceed the planet's gravitational potential.
Check the units:
3.
The escape velocity of Earth comes to about 11.2 kilometers per second from the
surface.
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Example
Calculate the escape velocity of the moon if Mass is 7.35 1022
kg and radius is 1.7 106 m.
Solution:
M = 7.35 1022 kg,
R = 1.7 106 m
Hence Escape Velocity is given by
So we could see that when the Apollo astronauts departed from the surface of the
Moon, they only had to be travelling one fifth the speed they travelled in order to
leave the Earth.
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Example
Calculate Mercury and Jupiter escape velocity.
Solution
Given:
Radius ( r ) = 2.4103 km = 2.4106 meters
Mass ( M )= 3.31023 kilograms
The escape velocity is given by:
Given:
Radius ( r ) = 7.15104 km = 7.15107 meters
Mass ( M )= 1.91027 kilograms
The escape velocity is given by
Answer:
Jupiter's escape velocity is (60/4.3) 14 times higher than Mercury's.
For Mercury
For
Jupiter
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A table of masses and radii is given below for many bodies in the Solar System.
Planet Mass (kg) Radius (km)
Mercury 3.301023 2439
Venus 4.871024 6051
Earth 5.981024 6378
Mars 6.421023 3393
Jupiter 1.901027 71492
Saturn 5.691026 60268
Uranus 8.681025 25559
Neptune 1.021026 24764
Example
Calculate the mass of the planet Mars, whose escape velocity is 5 103?
Given:
Radius of Mars (r) = 33.97 105 kg, Escape velocity (ve) = 5 103,
Mass of the planet mars (M) = ?
Substitute the given values in the formula:
Mars mass:
Answer: The mass of planet Mars is M = 6.3641023 kg
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LINKS:
www.smith-teach.com
www.planetfacts.org
www.chandra.harvard.edu
www.galaxymaine.com
www.astronomy.wonderhowto.com
www.universetoday.com
www.messenger-education.org
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