Maths and Astronomy Why Maths Comenius project

MATHS AND ASTRONOMY
G I M N A Z J U M I M . A N N Y W A Z Ó W N Y , G O L U B - D O B R Z Y Ń
MATHS AND
ASTRONOMY

THIS EBOOK WAS PREPARED
AS A PART OF THE COMENIUS PROJECT
WWWHHHYYY MMMAAATTTHHHSSS???
by the students and the teachers from:
GIMNAZJUM IM. ANNY WAZÓWNY
IN GOLUB-DOBRZYŃ IN POLAND
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.

II.. DDIISSTTAANNCCEE UUNNIITTSS IINN SSPPAACCEE
IIII.. HHOOWW TTOO CCAALLCCUULLAATTEE AACCCCEELLEERRAATTIIOONN DDUUEE TTOO
GGRRAAVVIITTYY??
IIIIII.. WWEEIIGGHHTT OONN OOTTHHEERR PPLLAANNEETTSS
IIVV.. HHOOWW FFAASSTT DDOO TTHHEE PPLLAANNEETTSS MMOOVVEE??
VV.. HHOOWW OOLLDD AARREE WWEE OONN OOTTHHEERR PPLLAANNEETTSS??
VVII.. HHOOWW TTOO CCAALLCCUULLAATTEE EESSCCAAPPEE VVEELLOOCCIITTYY??

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.
II .. DDD III SSS TTT AAA NNN CCC EEE UUU NNN III TTT SSS III NNN SSS PPP AAA CCC EEE
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.
Image from: www.mathscareers.org.uk
Image from:
Image from:

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
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
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?
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 1.141016 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
30 AU

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

II II .. HH OO WW TT OO CC AA LL CC UU LL AA TT EE AA CC CC EE LL EE RR AA TT II OO NN DD UU EE 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.

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 2.43106 m and its mass is 3.181023 kg.
Given:
 6

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

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?

II 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.

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.

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.

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%

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:

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.

II VV .. 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:

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

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.

VV .. 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

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:

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!

VV II .. 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 6.38106 meters and the
mass is 5.971024 kilograms.

We need the gravitational constant (G), which is 6.6710-11 . It is required to use
metric units for this equation:
Given:
Radius ( r ) = 6.38106 meters
Mass ( M )= 5.971024 kilograms
Gravitation constant = 6.6710-11
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.

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.

Example
Calculate Mercury and Jupiter escape velocity.
Solution
Given:
Radius ( r ) = 2.4103 km = 2.4106 meters
The escape velocity is given by:
Given:
Radius ( r ) = 7.15104 km = 7.15107 meters
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

A table of masses and radii is given below for many bodies in the Solar System.
Planet Mass (kg) Radius (km)
Mercury 3.301023 2439
Venus 4.871024 6051
Earth 5.981024 6378
Mars 6.421023 3393
Jupiter 1.901027 71492
Saturn 5.691026 60268
Uranus 8.681025 25559
Neptune 1.021026 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 = 6.3641023 kg

LINKS:
www.smith-teach.com
www.planetfacts.org
www.chandra.harvard.edu
www.galaxymaine.com
www.astronomy.wonderhowto.com

G I M N A Z J U M I M . A N N Y W A Z Ó W N Y , G O L U B - D O B R Z Y Ń

Maths and Astronomy Why Maths Comenius project

Recommandé

Recommandé

Contenu connexe

Tendances

Tendances (20)

Similaire à Maths and Astronomy Why Maths Comenius project

Similaire à Maths and Astronomy Why Maths Comenius project (20)

Plus de Gosia Garkowska

Plus de Gosia Garkowska (20)

Dernier

Dernier (20)

Maths and Astronomy Why Maths Comenius project