Celestial Mechanics

Celestial Mechanics
Celestial Mechanics
Orbital Maneuvers
Earth, Moon, Mars, and Beyond
Rodolfo G. Ngilangil Jr. MBA, BSAE
Impulsive Maneuvers
Impulsive maneuvers are those in which brief firings of onboard rocket motors change the
magnitude and direction of the velocity vector instantaneously. The position of the spacecraft is
considered to be fixed during the maneuvers. This is true for high–thrust rockets with burn times
short compared with the coasting of the spacecraft.
Orbital Maneuvers – 2
Hohmann Transfer – Definition
The Hohmann transfer is an elliptical orbit tangent to both circles at its apse line. The periapse and
apoapse of the transfer ellipse are the radii of the inner and outer circles.
The Hohmann transfer is the most ... Show more content on Helpwriting.net ...
Determine the most efficient transfer from orbit 1 to a circular orbit of altitude 16 000 km (orbit 3),
and the required delta–v.
Phasing Maneuvers
A phasing maneuver is a two–impulse Hohmann transfer from and back to the same orbit. Phasing
maneuvers are used to change the position of a S/C in its orbit.
Phasing Maneuvers – Assignment 2 :
S/C at A and B are in the same orbit 1. At the instant shown, the chaser vehicle at A executes a
phasing maneuver so as to catch the target S/C back at A after just one revolution of the chaser's
phasing orbit 2. What is the required total delta–v.

Phasing Maneuvers – ssignment 3 A
It is desired to shift the longitude of a GEO satellite from 99.1°W to 111.1 °W in three revolutions of
its phasing orbit. Calculate the delta–v requirement.
Non–Hohmann Transfers with A Common Apse Line – 1
Transfer between two coaxial elliptical orbits in which the transfer trajectory shar es the apse line
but is not necessarily tangent to either the initial or target orbit. The problem is to determi ne if there
exists such a trajectory joining points A and B.
Non–Hohmann Transfers with A Common Apse Line – 2
rA = rB = e3 =
2 h3
1 µ 1 + e3 cos θ A 1 µ 1 + e3 cos θ B
2 h3
rB − rA rA cos θ A − rB cos θ B cos θ A −
... Get more on HelpWriting.net ...

The Concept Of Transfer
by calculating the cost, instead, it determines the optimality using the NC [12]. The NC allows a
mission planner to analyze whether a transfer is optimum or non–optimum by evaluating the primer
vector pro le along the transfer trajectory and see if it satisfy the NC. Transfer trajectory refers to the
trajectory of the spacecraft from orbit around the departure planet to the orbit of around the
destination.
In case of non–optimum trajectory, one can alter the trajectory by modifying the orbital velocity
vector as such to satisfy the NC of optimality [13]. The modi cation of the velocity vector is
achieved through implementation of DSM. A linear method to optimize a transfer based on the
gradient of the cost functional was rst proposed by ... Show more content on Helpwriting.net ...
Solving primer vector equation using state transition matrix is the most common approach in the
trajectory optimization. Since this approach relies on the numerical integration of the STM and the
primer vector, the computation time becomes a disadvantage. With the rapid advancement in
computer technology, the issue of computation time is not as significant as in the past. However, the
numerical approach using STM did not give an insight to the optimization problem. In the paper by
Iorfida, it is presented how the analytical solution to the out of plane case can be explored to nd the
influence of the transfer orbit parameters to the optimality.
There are numerous works which have been conducted to derive the analytical solution to the primer
vector equation. Russel, in his paper, [25] presented an analytical solution to the primer vector
equation for the low thrust transfer case. His method is based on the analytical derivation of the state
transition matrix. Other work by Huang [26] also studied a low thrust transfer and proposed an
analytical solution for circular case transfer.
For an impulsive transfer case, there are several works, which focused on finding an ana–lytical
solution to the primer vector equation. In 1669, Prussing published his work on the analytical
solution of the out of plane component circular rendezvous problem [27]. In 1991, Carter published
his studies of the xed time linearised impulsive rendezvous

The Physics Of Satellites Orbiting Earth
Satellites orbiting earth will experience external perturbations that can affect the orbit. The primary
force acting on the satellite depends on the altitude of the orbit. The most significant effect is caused
by atmospheric drag which affects satellites in low earth orbit. Outside of low earth orbit, satellites
still experience external forces that could change the orbit however unlike atmospheric drag, the
orbit altitude will not necessarily decrease. In addition to atmospheric drag, satellites are subject to
thermal drag and gravity perturbations.
Atmospheric Drag: (Author: Emma)
Atmospheric drag effects all spacecraft orbiting earth below 2,000 km, producing both short
periodic and slow perturbations. It is the is the largest source of error in modeling the force on many
satellites because it is difficult to predict [2]. Atmospheric drag is caused by the collisions of gas
molecules with the spacecraft. Over time, the collisions start to decay the orbit. Demonstrating a
positive feedback effect, where drag decreases the orbital altitude where the density is greater which
produces more drag which then decays quicker. Factors that influence drag are the atmospheric drag
surrounding the spacecraft, as well as the spacecrafts drag coefficient, area, mass and velocity [1].
The atmospheric density is not constant and changes with solar activity. The solar magnetic activity
cycle is the periodic change in the sun 's activity that lasts around 11 years. During solar maximum,

Interplanetary Dust and Orbital Debris
INTERPLANETARY DUST AND ORIBITAL DEBRIS
Introduction
The interplanetary dust particles are sized in between few cm to molecules. The dust particles are
charged with +ve charge due to solar wind. The positive charge helps in reacting to the Lorentz
force. The interplanetary dust grain population mainly originates in the main asteroid belt that is
located between 2 to 4 astronomical units (AU) from the Sun. Dust forms in collisions between
bodies of diﬀerent sizes and energies or may be carried in by comets. 10 % of the dust formed by
collisions in this area is transported towards the Sun via the Poynting–Robertson drag. The
remaining 90 % leave the solar system hyperbolic orbits as newly formed interstellar dust. Dust
grain orbits are mainly inﬂuenced by gravity. Their orbital motion is Keplerian. Most of the dust
grains stemming from the asteroid belt orbit the Sun with low eccentricities and, thus, near–circular
orbits.
Material properties of IDP
The Most of the dust in the inner solar system is assumed to originate from parent bodies orbiting in
the Main Asteroid belt .the materials are divided into 3 main Chrondrites 60%,Achrondrites 10%
and Irons 30%. Chrondrites contains Mg, Al, Si, S, Ca, Cr, Mn, Fe and Ni.
IDP motion dynamics
The dynamics of interplanetary dust particles are determined by a variety of forces that
Can change the parameters of a dust grain's orbital motion. Very small IDP has perturbation by other
forces. The P–R effect (pointing–Robertson effect)

Essay on Johannes Kepler
Johannes Kepler was a German astronomer and mathematician who lived between 1671–1630.
Kepler was a Copernican and initially believed that planets should follow perfectly circular orbits
("Johan Kepler" 1). During this time period, Ptolemy's geocentric theory of the solar system was
accepted. Ptolemy's theory stated that Earth is at the center of the universe and stationary; closest to
Earth is the Moon, and beyond it, expanding towards the outside, are Mercury, Venus, and the Sun
in a straight line, followed by Mars, Jupiter, Saturn, and the "fixed stars". The Ptolemaic system
explained the numerous observed motions of the planets as having small spherical orbits called
epicycles ("Astronomy" 2). Kepler is best known for introducing three ... Show more content on
Helpwriting.net ...
His first law states, "The orbits of the planets are ellipses, with the Sun at one focus of the ellipse."
As shown in Figure 1, The Sun is not at the focus of the ellipse, but is instead at one focus [usually
there is nothing at the other focus of the ellipse]. The planet then trails the ellipse in its orbit, which
implies that the Earth–Sun distance is continually changing as the planet goes around its orbit.
Kepler's second law states, "The line joining the planet to the Sun sweeps out equal areas in equal
times as the planet travels around the ellipse." As shown in Figure 2, an imaginary line from the
center of the sun to the center of a planet sweeps out the same area in a given time. This means that
planets move faster when they are closer to the sun. Kepler's third and final law states, "The time
taken by a planet to make one complete trip around the sun is its period. The ratio of the squares of
periodic times for two planets is equal to the ratio of the cubes of their mean distances from the
sun." Kepler's third law indicates that the time taken by a planet to orbit the Sun increases quickly
with the radius of its orbit ("Johannes Kepler: The" 1–4). Kepler's laws challenged Aristotelean and
Ptolemaic astronomy. His statement that the Earth

The Solar System
1.1. Natural satellites
Formally, classified natural satellites or moons include 176 planetary satellites orbiting six of the
eight planets. Of the inner planets, Mercury and Venus have no natural satellites; Earth has one large
natural satellite, known as the Moon; and Mars has two tiny natural satellites, Phobos and Deimos.
The large gas giant planets have extensive systems of natural satellites, including half a dozen
comparable in size to Earth 's Moon. The objects in the solar system are classified foremost by their
dynamics and motion, while size and composition are important for secondary classification. The
solar system itself is, defined by the orbits of its constituent bodies. Classification by motion is more
useful for astronomers. In fact, the dynamics of the solar system were a central issue for early
astronomers such as Ptolemy in the second century and Nicholaus Copernicus in the 16th century. In
what follows, we introduce different classifications of the planetary moons.
1.2. Classification of motion of natural satellites
1.2.1 Classification based on the orbits
The natural satellites may be, classified into three categories according to their orbits (Newburn and
Gulkis, 1973; Burns, 1986) as follows.
Category 1: Regular satellites
The regular satellites move prograde in nearly circular orbits in the equatorial plane of the mother
planet. This sector of satellites represents the four Galilian satellites of Jupiter, the eight classical
satellites of Saturn

Horinzontal
Chapter 3
Falling Objects and
Projectile Motion
Gravity influences motion in a particular way.
How does a dropped object behave?
!Does the object accelerate, or is the speed constant?
!Do two objects behave differently if they have:
!different masses?
!different shapes?
Acceleration Due to
Gravity
" Earth exerts a gravitational force on objects
that is attractive (towards Earth's surface).
" Near Earth's surface, this force produces a constant acceleration downward.
#
#
#
To measure this acceleration, we need to slow down the action.
Galileo was the first to accurately measure this acceleration due to gravity.
By rolling objects down an inclined plane, he slowed the motion enough to establish that ... Show
more content on Helpwriting.net ...
The velocity–versus–time graph for a certain falling object is shown.
Is the acceleration of this object constant? a)
b)
c)
Yes.
No.
Impossible to tell from this graph.

Constant acceleration would require the v vs. t curve to be a straight line. This graph is curving
upward, so the slope
(and the acceleration) is increasing. Throwing a ball downward " Let the ball be thrown downward
instead of being dropped.
It will have a starting velocity different from zero. # It will reach the ground more rapidly.
# It will have a larger velocity when it reaches the ground. v = v 0 + at
#
1 2 d = v 0 t + at
2
Beyond Free Fall:
Throwing a Ball Upward
! What if the ball is thrown
upward?
Gravitational acceleration is always directed downward, toward the center of the Earth.
! Here, the acceleration is in the opposite direction to the original upward velocity.
!
" Let the initial velocity be 20
m/s upward.
#
#
It immediately starts experiencing a downward acceleration due to gravity, of approximately 10 m/s.
Every second, the velocity decreases by 10 m/s.
" After 2 s, the ball has reached
its highest point.
#
Its velocity changes direction, from upward to downward, passing through a value of 0 m/s. " Now,
the downward
acceleration increases the downward velocity.
What is the ball's acceleration at the top of its

Space Junk Epidemic
Space Junk Today, speeding past our Earth ten times faster than a bullet (tweentribune), are millions
of pieces of space junk ranging in size from flecks of paint to entire Satellites (NASA). To most
people this does not sound like much of a problem, but even a fleck of paint moving 17,500 miles
per hour or more can cause significant damage to spacecraft and space walkers alike. As more
collisions take place and as new items enter the atmosphere the statistical probability of a collision
will increase. Eventually, the probability of a collision will become so high that it will be unsafe to
enter space from Earth (NASA). Now, as discussion of colonizing Mars intensifies, space junk has a
new way of spreading and clogging up more of the universe. When humans colonize Mars they will
... Show more content on Helpwriting.net ...
As of December 2015 there were 1071 operational satellites around the Earth and over 21.000
objects larger than 10cm hurtling around the planet (Universetoday). In 2013 there were 81 space
launches and in 2014 that number increased to 92 space launches. 2014 had the highest number of
space launches in a year in more than two decades (spaceflightnow). If those on Mars could cut back
on the number of Satellites they put up they would have a significantly lower statistical probability
of collisions. This would make long term space use possible.
Another important anti–space junk component is banishing pieces of rockets that fall off, like solid
rocket boosters on the Space Shuttle (Space.com). These large chunks of debris would cause
extreme amounts of damage. If a fleck of paint can harm satellites and space suits then a 177 foot
long, 12 foot diameter, 1.6 million pound Solid Rocket Booster would be cataclysmic. If various
space organizations can reduce the number of rocket stages they use, they can dramatically reduce
the amount of large space junk orbiting the Red

How Does GPS Use Trilateration To Pinpoint An Unknown...
This simplified 2D example will demonstrate how GPS uses trilateration to pinpoint an unknown
location. Once a receiver has "locked in" onto a satellite, the unknown position may be anywhere
within the radius of 1000 km of satellite one. The receiver picks up a signal from a second satellite
and calculates the range as 1200 km. The position has been narrowed down to two positions; where
the ranges intersect. If the receiver picks up another satellite, the unknown position can be found by
calculating where all three distances are possible.
The Global Positioning System (GPS) is a constellation of twenty four satellites that can pinpoint
any position on earth using a GPS receiver. Used to calculate exact position, speed and time, there
are tremendous ... Show more content on Helpwriting.net ...
Many accidents are caused by distracted drivers who are attempting to punch in the address while
driving. Making sure that you have added the address correctly and knowing the general route
before driving is the first step to proper use of your GPS device. Take advantage of GPS systems
that use voice. Using this feature helps one to keep their eyes on the road, avoiding unnecessary
distraction by looking at the GPS device. Another safety tip is to put your passenger in charge of the
directions. A passenger can be used as an alternative to voice directions, so one can keep their eyes
on the road. Understand there may be changes not reflected on the GPS information. It is important
to pay attention to the roadway and not depending on the device because maps may not be
updated.It is important to understand how to operate your GPS device and to take advantage of
safety features such as voice control. YouTube has many videos that may walk one through the steps
to understand how to use your specific GPS.The principal rule of safety when using the GPS device
is to always keep your eyes and the road and not your device. Using the GPS device can be a great
tool to get you to your destination safely, especially when used

Being Killed In Space
There are four common ways for a human to be killed in space.
Loss of life support. If a human being is not kept warm, nicely pressurized and fed with specific
chemicals and oxygen, death comes quickly. Kinetic energy. Whether that means a thousand tons of
rock plowing through a body at a few hundred kilometers per second or neutrons near light speed is
almost irrelevant. How fine–grained the kinetic energy is makes just a minuscule difference. The
human is dead anyway.
Electromagnetic radiation. The particle aspect of photons aside, this has an entirely different quality
than kinetic energy above. Being slow–cooked to death is decidedly more unpleasant than just being
blown to smithereens.
And then there are singularities. If whatever happens to a human makes it impossible to interact
with the rest of universe in a meaningful way, the human might as well be dead. Whoever the
human's loved ones are, they will lose contact forever. If the human doesn't have loved ones, this
might not be the worst way to go.
Kim had few things to worry about at the moment. Those few, however, were sources of great
concern. She was in a system called Relay where the largest physical threat was a singularity which
she needed to stay ... Show more content on Helpwriting.net ...
This would have been much easier if it hadn't been for the pair of problems less than a light second
away. Those actually followed her around. _Much like many of my other problems_, she thought to
herself. These two ships had followed her to Relay despite the system not being used for FTL traffic
anymore.
Their vectors intersected with her current heading in a rather displeasing way. They weren't out to
kill her, it seemed, since they didn't use any weapons. But they were still very insistent, probably
keen on boarding her ship. Some would have argued it wasn't _exactly_ hers, which was at the very
heart of why yet a different and

A Short Note On Mission And Mission Of Mars
Wei Jiang
ESS 102
Instructor: Erika Harnett
November 18, 2016
Manned Mission to Mars The year is 2045, a time when the earth is in great turmoil. Tsunamis,
hurricanes, tornadoes, droughts, famines... all the impacts of global warming are just starting to
disrupt life. Europe and North America are experiencing immigration in unprecedented numbers,
neither because the migrants are escaping persecution, conflicts and wars in their countries, nor
because they are pursuing economic prospects; but because they can no longer survive the high
temperatures in their countries. Europe and North America have become too warm for their
inhabitants but just the right temperature for migrants from South America, the Middle East, Africa
and South Asia to survive. But the problem is not just global warming and migration – there is a
more frightening threat, one that could turn the entire Earth's crust upside down – Nuclear War. All
political and diplomatic negotiations have failed, leaving war as the last resort. There is not a single
safe spot on earth to escape to. There is only one option left –a place outside earth, a different planet
– Mars. The possibility of relocation to Mars comes as big relief to people on earth, but there is one
problem – not everybody can go. The mission is too expensive for ordinary people. Only billionaires
and politicians who have stolen (and are still stealing) funds that belong to their states can afford it.
However, the good news for a few lucky people is

Johannes Kepler Essay
Johannes Kepler
Johannes Kepler is now remembered for discovering the three laws of planetary motion, and writing
about them in books that were published in 1609 and 1619. He also did important work in optics,
discovered two new regular polyhedra, gave the first mathematical treatment of close packing of
equal spheres, gave the first proof of how logarithms worked, and devised a method of finding the
volumes of solids of revolution. This can be seen as contributing to the development of calculus.
Not only did he help the development of calculus, but he calculated the most exact astronomical
tables known today. This accuracy did much to establish the truth of heliocentric astronomy, which
states that ... Show more content on Helpwriting.net ...
The curriculum included geocentric astronomy. This is the study of how all seven planets at the time
– Moon, Mercury, Venus, Sun, Mars, Jupiter and Saturn – moved around the Earth. Their positions
to the stars were being calculated by combining circular motions. This system was what worked
with the current physics of the time, although there were certain difficulties. However, these
astronomers, who also saw themselves as mathematicians, were content to carry on calculating
positions of planets. They figured they would leave it to natural philosophers to worry about
whether the mathematical figures added up to the physical ones. Kepler, however, did not take like
this attitude. His earliest published work in 1956 says that people should consider the actual paths of
the planets, and not just the circles used to construct them.
At Tubingen, Kepler studied Greek and Hebrew as well as mathematics. Ironically, at the end of his
first year Kepler got A's in every subject except mathematics. It is supposed that Maestlin was trying
to tell him he could do better, because Kepler was one of the select pupils that Maestlin chose to
teach more advanced astronomy to. He was teaching them the new, heliocentric cosmological
system that was formed by Copernicus. Kepler almost instantly accepted that the Copernican system
was physically true.
It seems that even in Kepler's student days there were signs that his religious beliefs were not

How Aristarchus Made A Large Jump From Terrestrial...
How Aristarchus made a large jump from terrestrial measurements of scale to the celestial is truly
impressive. Without any measures of the sizes of and distances to any celestial objects, Aristarchus
was able to measure both for the Moon. He was able to use something he knew about the Earth and
a bit of smart geometry to deduce the size of the Earth 's shadow at the distance of the Moon. By
watching the Moon go through the Earth 's shadow which is formally known as a lunar eclipse,
Aristarchus could determine the ratio between the size of the shadow at the distance to the Moon
and the size of the Moon.
Aristarchus knew that on average two times a year, the surface of the full Moon becomes dark for a
period ranging up to around two hours. It was reasoned that during these times, the Moon was
passing through the shadow cast by the Earth, known as a lunar eclipse. The eclipse would only
occur during the full Moon because that 's the time when the Moon is on the opposite side of the
Earth from the Sun. Aristarchus realized that the longest lunar eclipses must occur when the Moon
passes right through the center of the Earth 's shadow, and so that the angular size of the Earth 's
shadow could be measured by noting the difference in the position of the Moon before and after the
eclipse. He could then compare this angular size to the angular size of the Moon to get the ratio
between them. He used the formula Now by using the formula.
The moon could fit into earth roughly

Satellite Motion Lab Answers
Dana Chernysheva
1A
Satellite Motion Lab
1.8)
1.9) A circular orbit means that the magnitude of velocity is constant. In a circular orbit, the radius is
the same at all points on the orbit. If the radius is the same, then the object is always the same
distance away from the central body, so the gravitational pull is the same because a= v^2/r, and v
and r are the same at any given point. An elliptical orbit would have changing velocities at certain
points. An ellipse does not have one constant radius. This means that the object comes closer to the
central body at certain points and further away at other points as it travels along the orbit; when it
gets closer to the central body the object goes faster because there is a greater gravitational ... Show
The Earth revolves around the sun due to the gravitational force between them, so the Earth is pulled
by the gravitation between it and the Sun. The Sun moves in an orbit as well due to the same
gravitational force between it and the planet, which is the same as in real life. Also, the Moon
revolves around the Earth like the moon revolves around the planet. It can be observed that the
moon moves faster when the planet (Earth) is closer to the sun. This can be confirmed using
Kepler's Second Law (Equal Areas) because it states that the speed of the planet constantly changes
and becomes faster when it is closer to the sun because it is in an elliptical

What is a satellite? How can they help us daily / what is...
What is a satellite?
A satellite is a small thing orbiting or circling a larger thing. The complete path it follows is called
an orbit. The moon is a example of a natural satellite of the earth. Manmade, or artificial satellites
are placed into orbit by rockets or space shuttles.
After World War II, the former Soviet Union successfully launched Sputnik I, the first artificial
satellite in 1951, into space. In 1958, the United States launched its first artificial satellite Telstar I
into orbit. Since then, many more satellites were made by different countries and launched into
space.
How can they help us daily / what is the purpose with satellites
It has been helping scientists find answers to the unknown, and assisting tourists finding ... Show
The closest point a satellite comes to the Earth is called its perigee. The farthest point is the apogee.
The time it takes a satellite to make one full orbit is called its period. The slope is the angle the
orbital plane makes when compared with the Earth's equator.
Isaac Newton's First Law of Motion sounds like this any given object there is in motion will stay in
motion unless something pushes or pulls on it. Without gravity an Earth–orbiting satellite would go
off into space along a straight line. With gravity it is pulled back toward the Earth. There is a
constant tug–of–war between the satellites tendency to move in a straight line, or momentum, and
the tug of gravity pulling it back.
An object's momentum and gravity have to be balanced for an orbit to happen. If the forward
momentum of one object is too great, it will speed past the other one and not enter into orbit. If
momentum is too small, the object will be pulled into the other one and crash. When these forces are
balanced, the object is always falling into the planet, but because it's moving sideways fast enough,
it never hits the planet.
Escape velocity is the speed an object must go to break free from a planet?s gravity and enter into
orbit. Escape velocity depends on the mass of the planet. Each planet has a different escape velocity.
The object?s distance from the planet?s center is also important. The escape velocity from the Earth
is

Rough Guide To Exoplanets
Alex's Rough Guide to Exoplanets
Intro
By exploring outer space, astronomers have found distant planets, called exoplanets. Properties of
these planets can be determined by using various techniques and technology.
The definition of an exoplanet is a planet that orbits a star outside our solar system. The first
exoplanet was discovered in 1995.
Nowadays techniques are becoming more refined and by using them predictions can be made based
on the many data points that we have for these planets. For example the presence of certain elements
can be detected via spectroscopy and then that data can be used to aid a prediction into a planet's
climate. Because there are so many factors that determine properties such as climate the more data
gathered ... Show more content on Helpwriting.net ...
This is because for a planet to be detected it must have an orbit that transits between the observation
point and the star. This means that the planet must orbit in a very specific plane and that the plane
becomes smaller for planets that orbit at a greater distance from their host star. So planets that are
detected using the transit method are usually planets known as "Hot Jupiters". To break down this
description it means that these planets are comparable to Jupiter because of their size and density,
and are described as being hot due to their closeness to their host star (usually between 0.015 and
0.05 AU) and the high surface temperature that comes with that closeness. Also if the host star is
small such as a neutron star then the size of the plane that a transit is visible in is also reduced.
Another drawback is that the amount of false positive candidates is higher than many other methods
and it's thought that, in some cases, as many as 35% of detected planets from Kepler are false. This
means that in some cases follow up investigation is required before a planet can be confirmed. A
common source of false positives is caused by fluctuations in the star's brightness which are not
caused due to a transiting planet but are often hard to differentiate and so they can cause false
negatives

Vertical Velocity Lab Report
What we changed:
To decrease the time to fall, we maximized the launch speed while simultaneously reducing the
height to a minimum and launching from an angle of 0 degrees. The time a projectile spends in the
air is directly related to the angle measurement and height. The greater either of the two variables,
the more hang time a projectile would experience. Therefore, to reduce time, our group minimized
both variables, effectively rendering time from increasing. Furthermore, our group increased the
launch speed because a faster velocity meant that it would reach the ground faster in less time.
However, speed is relative, so we set the conditions to allow speed act on a shorter distance,
yielding less time.
What we observed:
In the first ... Show more content on Helpwriting.net ...
However, the vertical velocity (vy) is changing, because the object's height changes throughout the
entire movement. This is because it is thrown at an angle, and gravity works against it to bring it
back to the ground.
2. Carefully observe the vertical velocity vector as the projectile approaches the highest point (i.e.,
the peak). At its highest point, the vertical velocity is ______. At its peak, the vertical velocity is 0.
This is because gravity is pushing against the force of the throw. Gravity fights to keep the object on
the ground. At the peak of the object's height, the force of the throw and gravity are equal.
3. Use the arrows to measure the time it takes to rise to the peak, the total time in the air, and the
time it takes to fall from the peak. Record to the first decimal place. The time rising is about 5.3
seconds, the time falling is about 5.3 seconds, and the total time in the air is about 10.6 seconds. All
three of these measurements have been rounded to the 10th place. This shows that the amount of
time that an object takes rising is the same amount of time the object takes falling, when there is no
air resistance, of course. This also means that with no air resistance, the time it takes to rise doubled
is the total time in the

Moons Of The Terrestrial Planets
Moons of the Terrestrial Planets
2.2. Moons of Mars
2.2.1 Introduction
Mars orbits at a distance of 1.52 AU from the Sun. Mars has two natural satellites, discovered by
Asaph Hall in 1877. The innermost of these, Phobos, is about 22.2 km in diameter (27.0×21.6×
18.8) and orbits the planet with a period far less than Mars 's period of rotation 7.7 hr. (only one
quarter that of Deimos), causing it to rise in the west and set in the east(Born & Duxbury 1975).
Thus, it makes more than 3 orbits in a single Martian day. As it orbits, it slowly spirals in towards
the Martian surface. Phobos is drawing closer to Mars by 2 meters every one hundred years, and it is
predicted that it will collide with the planet /or destroyed in the atmosphere. Phobos has a semi–
major axis 9376 km (2.76 Mars radii), eccentricity 0.0152 and orbit mean inclination 1.1o (relative
to Mars' equator).
The outer satellite, Deimos, is about 12.6 km in diameter (15.0×12.2× 11.0) and orbital period about
30 hr. In fact, Deimos is the smallest known moon in the Solar System. Like our own Moon, Deimos
orbits far enough away from Mars that it is being, slowly pushed farther and farther away from the
planet. Deimos has a semi–major axis 23463.2 km (6.92 Mars radii), eccentricity 0.0003 and orbit
mean inclination 2.4o (relative to Mars' equator).
Both Deimos and Phobos have very circular orbits, which lie almost exactly in Mars 's equatorial
plane. They are both small, irregular objects comparable in size to the

Orbital Parameters Of Earth And Mars
Question 5
It is assumed that Earth and Mars are in Circular orbits. The initial relative phase angle is stated to
be 30o. Calculate the transfer orbit of a time of flight (TOF) 120 day. The orbital parameters such as
ΔVs must be calculated. In addition, the propulsion considerations must be evaluated for the case
when mission is to be Evaluate the orbital parameters, ΔVs and propulsion parameters when the
mission is to be executed by a 500 kg spacecraft, which utilizes 450 s specific impulse propellant.
Assumptions:
Orbits are coplanar. Initial position of earth is taken as reference direction and epoch of the mission
Data:
2The total Time period of Earth = 365 days
1Hence, the time period of Mars = 365 x 1.881 = 686.565 days
3Distance of Earth from Sun = 149.5 x 106 kilometers
4Distance of Mars from Sun = 227.8 x 106 kilometers
5Initial position of planets: 7After 120 days position of planets:
8Change in position of Earth = 120/365 = 0.32876 = 32.876% of the complete orbit.
Change in angular position of Earth = 0.32876 x 360o = 118.356o
Final position of Earth with reference direction = 118.356 + 0 = 118.356o
Change in position of Mars = 120/686.56 = 0.17478 = 17.478% 0f the complete orbit.
Change in angular position of Mars = 0.17478 x 360o = 62.9219o
Final position of Mars with reference direction = 62.9219 + 30 = 92.9219o Transfer ellipse:
Now the transfer ellipse will be such that initial position of Earth will be the point of launch and
perigee of the

Non Gravitational Effects On Jupiter 's Satellites
Non–gravitational effects on Jupiter's satellites The non–gravitational forces affecting on the natural
satellites orbits were studied recently using observations. This effect creates a quadratic term in
orbital longitude and a secular changing of the semi–major axis. Causes may be only hypothetical. It
is assumed that the causes of the effects are in the tides in the body of the planet and in the body of
satellite (Emelyanov 2015; private communication). Lainey et al. (2009) considered an extensive set
of astrometric observations from 1891 to 2007 to study the orbital tidal evolution of the innermost
Jovian satellite Io. They integrated numerically the full equations of motion for the satellite center's
mass. It is noteworthy that Io's tidal heat comes from the orbital energy of the Io–Jupiter system
(resulting in orbital acceleration), whereas dissipation of energy in Jupiter causes Io's orbital motion
to declare. They used a weighted least squares inversion procedure and minimized the differences
between the observed and computed positions of the satellites to determine the parameters of the
model. The solution for the tidal dissipation gives for Io, and for Jupiter, where, is the love number
and is the quality value. The orbital accelerations represent a shift in the orbital positions of the
Galilean satellites Io, Europa and Ganymede of 55 km, –125 km and –365 km over the 116 years
respectively. The fit astrometric residuals range essentially between 0.02 and

Solar System and Gravity
The Effects of Gravity
There are some people who worry that when they 're outside, if they don 't keep a
good grip on the ground, they 'll just go flinging off into space. They needn 't really worry
about this, because gravity generally keeps that sort of thing from happening. The thing is,
no one is really sure what causes gravity, but the effects have been studied by many
physicists and astronomers. Three of the more obvious effects of gravity are things falling
down, weight, and the the moon and planets staying in their orbits.
Things fall down. People have generally grown to accept that if one lets go of
one 's prized and valuable textbook when walking through a mud puddle, the book will
invariably ... Show more content on Helpwriting.net ...
So, too, the moon continuously falls around the Earth, and
the planets fall around the Sun. Each of these objects balances the Sun 's force of gravity
with its own momentum, that is, with its innate tendency to keep moving in the same
direction at the same speed. Take away the Sun 's force of gravity, and the planets would
sail into space, in whatever dircetion at whatever speed they happen to have. Take away a
planet 's momentum, perhaps with a giant hand that stops it in orbit, and it would head
straight for the Sun, drawn by the Sun 's gravitational force. But, because both gravity and
momentum are at work, the planets "fall", not down, but around, forever as they move in

elliptical, but nearly circular, orbits.
Working with insight that has never been exceeded since, Isaac Newton showed
that by making one simple assumption, that all objects in the universe attract one another
in accordance with a simple relationship, he could explain the fall of objects to the ground,
the moon 's orbit around the Earth, and the planets ' motions around the Sun. The
relationship Newton conceived was this: Every object attracts every other object with an
amount of force that varies in proportion to the product of the two objects ' masses, and in
inverse proportion to the square of the distance between the centers of the objects. It is
plain to see that gravity is quite important. The

Personal Narrative-Elmar Relationship
"Alright I am going to try and call Mom and Dad because they probably wonder how me and Elmar
are doing." I thought to myself as I dialed the phone and pressed the flashing green call button.
"Beep, Beep," The phone went to its recording "You may leave a message if you like, or press one
for more options" "Mom, Dad, are you there, I know you are wondering where Elmar and me are.
Well we are in space in a homemade rocket. Well you were gone, we built a little rocket. The rocket
is really small It has two beds, a bathroom, a fridge, and four buttons that were supposed to help us
control the rocket. The solar system has so many stars It is really cool. I did not expect it to be able
to make it into space. Me and Elmar are trying to find a way to make it back home. So if you get this
message you know where we currently are, bye." I pressed the red phone button and the call went
dead and I was sitting there hoping they would get the message. ... Show more content on
Helpwriting.net ...
I am 13 years old and my brother is 11. Right after my parents left for vacation for two weeks, we
sat down and watched a show of were two kids who were the first kids to ever go into space. Me and
Elmar (who is my younger brother) decided it would be cool to go into space so we decided to build
a rocket ship that would hopefully make it out of the atmosphere. By the time the show was was
done it was 10 o'clock and we decided to go to sleep. The whole night I could not sleep because I
was thinking of how we could make a really good

Accomplishments Of Johannes Kepler
Abstract
Born in a poor family, Johannes Kepler experienced the trauma of losing his father at a young age.
Facing confrontations with the Church and religious sects, he was forced to leave his home and
move to Prague. There, he gained the opportunity to work with a renowned Danish astronomer,
Tycho Brahe. Inheriting the precise data Brahe collected after Brahe died, Kepler delineated his
most famous discoveries––his three laws of planetary motion. Other than those three laws, he also
made great contributions to the scientific field: he made discoveries in optics, gave explanations to
the cause of tides on Earth, and invented logarithm.
Keywords: three laws of planetary motion, elliptical orbit, calculation
Accomplishments of Johannes Kepler ... Show more content on Helpwriting.net ...
He was quite unfortunate when he was young. He lost his father, who died in a battle in Holland,
when he was five, and he was left to be taken care of solely by his mother. In addition, he had bad
health in his childhood due to a disease called smallpox. However, even though he was physically
ill, he was mentally strong. His astonishing problem solving skill was shown when guests at his
grandfather's inn asked him to solve problems involving numbers. What is more, under the influence
of her mother's love of the natural world, Kepler had the opportunity to get exposed to stars, comets,
and other objects in the sky at night. His mother had helped to shape her son's path to explore the
unknowns and to change people's perceptions on the solar system and the

Parallax, Star Brightness, And Kepler's Laws Of Planetary...
I chose Proportional Scales in the Solar System, Examining Parallax, Star brightness, and Kepler's
Laws of Planetary Motion because I understood each section better than the other ones. Honestly I
had some corrections to make in each assignment but in the end they were almost perfect. My first
reaction was to choose topics that were the easiest to write about. But I thought about it and I chose
the topics that I best understood. I enjoyed these activities the most and if I had to choose my
favorite thing I've ever learned in Science a couple of those would be on the top of my list.
I chose the four assignments because I did my best job on them. Proportional Scales in the Solar
System deals with proportionality, obviously. The first thing ... Show more content on
Helpwriting.net ...
The biggest thing that stands out to me when I read that question is the Parallax assignment. When
we started that assignment I was so confused, and lost I just didn't understand it. But when we were
done with it I took the most out of that assignment than any other one. My first example is the
answer to the question we answered at the beginning, and I had no idea what to put so I answered
with one word, again hoping out of pure luck I would be right. If you were to ask me "what is
parallax?" I would probably stare blankly with no idea what it was, and at the end of the assignment
I could tell you almost anything about it. My second example is how well my parallax summary
turned out, and even my graph. I completely understand how when the distance to an object
increases the observed parallax decreases. I think the moment I understood everything was when I
held up my thumb straight out in front of me, closed one eye and then the next. I know, it sounds
dumb but that intrigued me into the assignment, when a little part of me finally understood it. My
notebook helped me have everything right in front of me opposed to searching for multiple papers.
Being a visual learner scared me going into high school until I had the opportunity to see everything
at once. In all honesty the one thing that would improve my notebook would be motivation to
myself. If I don't want to answer those questions we were assigned to do, or even write a summary
that

Kepler 's Laws Of Planetary Motion
Kepler's Laws of Planetary Motion Johannes Kepler was a German mathematician and astronomer.
Best known for his laws of planetary motion, Kepler was a key figure in the scientific revolution in
the 17th century. His laws illustrate the fundamental properties of the orbits of the planets and were
also a precursor to the formulation of Newton's law of gravitation. Biography: Kepler was
introduced to astronomy in his early years and developed a passion for it, which would span his
entire life. At age six, he observed the Great Comet of 1577. At age nine, he observed another
astronomical event, a lunar eclipse in 1580. While in school, Kepler had proven to be an excellent
mathematician and earned a reputation as a skillful astrologer, being able to cast horoscopes for
fellow students. He also learnt both the Ptolemaic system and the Copernican system of planetary
motion and, similar to Copernicus, believed that the sun was the centre of the solar system. He
became a mathematics and astronomy teacher at a school in Graz, despite his desire to become a
minister, but soon left to Prague and met Tycho Brahe. Tycho Brahe: Brahe, from a rich Danish
noble family, was fascinated by the vast solar system and the universe. Disappointed with the
accuracy of the data for the motions of the planets at the time, he decided to dedicate his life and
resources to recording planetary positions, which proved to be far more accurate than the best
previous work. After some early successes, and

Nt1310 Unit 9 Lab Report
Go to http://phet.colorado.edu/simulations/sims.php?sim=My_Solar_System and click on Run Now.
1. Move the slider all the way to accurate, click on the tape measure and the grid.
2. Click the radio button for 4 objects and run the simulation until the purple planet (body 2) has
made one complete orbit (one year).
3. After the first orbit (year), turn off the traces (show traces box) and watch another orbit (year) of
the purple planet (body 2).
Question One:
Is blue moon (body 3) circling the yellow sun (body 1) or the purple planet (body 2)? Explain your
answer.
Using the trojan asteroids setting it looks like all 3 bodys are orbiting the yellow sun
4. Increase the mass of the sun (body 1) to 400 and allow the simulation to run ... Show more
content on Helpwriting.net ...
Why?
When the mass of the sun is 1000 and the planet is 10. Because the mass of the sun is 10x larger
than the planet.
Question Five:
What makes the length of the year increase and decrease? Why?
The closer to the sun a planet is the shorter the year and the farther away it is the longer the year.
Question Six:
A planet in a circular orbit would always be the same distance from the sun. What do you notice
about orbits with the shortest years? Why? Shorter year = closer to the sun. The closer it is to the sun
the more of a pull the gravity has on it which causes it to orbit quicker.
8. Choose the ellipses preset from the pull–down menu.
9.You may move the slider bar about 2/3 of the way towards fast for this simulation.
10. Run the simulation until the green planet (body 4) returns to its starting point (one planetary
year)
Planet Time of One Orbit (planetary year) Closest Distance to Sun (perihelion) Farthest Distance to
Sun (aphelion)
Purple Planet (body 2) 2.2s 54 84
Blue Planet (body 3) 7.6s 54 250
Green Planet (body 4) 14.7s 54 419
11. Change the y velocity of the blue planet (body 3) to 90 and the green planet (body 4) to 70.
12. Run the simulation again until the green

Centers Of Gravity Summary
Centers of Gravity Most of the issues that arise while identifying centers of gravity concern the
ambiguity and multiple definitions of the term. Dr. Strange contends that modern military
definitions of centers of gravity (from Joint Publications) differ from Clausewitz's original
definition. The author further offers that due to the confusion from varying definitions, military
planners waste hours of debate deciding upon the center of gravity; time that could have been better
spent on planning. Even when using the simplest of definitions, equating centers of gravity to where
mass in concentrated and is the most appropriate target, issues can arise from identifying centers of
gravity of an adversary engaged in irregular warfare. Trying to identify centers of gravity (from a
counterinsurgent's point of view) is a difficult endeavor when faced with adaptive, evolving
insurgent groups. For example, a movement may begin with a dynamic core group and evolve into a
massive social movement that incorporates the populace. A decisive strike may eliminate an
assumed center of gravity but the ... Show more content on Helpwriting.net ...
Strange. Moral centers of gravity revolve around people and can be broken into three categories:
leaders, the ruling elite, and a strong–willed population. The author simplifies these three categories
by asking certain questions to determine true vs. false moral centers of gravity. The answers will
confirm or deny who is the actual moral center of gravity. For instance, if a movement or group
loses its will or direction by the loss of a leader, the leader is the moral center of gravity. If the ruling
elite can remove a leader, can replace them at will, and make actual decisions for the entity, then
they are the moral center of gravity. Lastly, if the will of the population is the driving force, then the
leadership may not be as important to the overall

Space Debris
Richard Xu
ESS102 AD
4/13/17
Research Paper RD
Topic: Space Debris
As Earth becomes more technologically innovated and crowded, space has become the answer of
many of our problems. Over the past couple of decades, we have launched numerous contraptions
into space such as satellites in which many currently orbit the Earth. These contraptions can be used
for reconnaissance, communications, weather, search and rescue, navigation, exploration, television,
and much more. Not a day passes by where we do not make use of these contraptions in our lives.
However, as we progress towards the future, more and more of these satellites will be launched into
orbit and as the fate of technology, they are bound to fail at some point and often get abandoned. ...
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Despite the rapidly increasing amount of space junk, planning of the clean–up of this junk is
underway.
Every year, the amount of debris orbiting the Earth grows, creating an increasingly hazardous zone
for satellites and even astronauts. Although some objects or shrapnel look small, their destructive
force is greatly magnified by the hostile conditions of space. According to NASA, pieces as small as
softballs can be traveling at an astonishing 17,500 miles per hour. Due to these enormous orbital
speeds, any of these objects have the potential to damage or destroy the satellites that we depend on.
In fact, even a tiny paint fleck has caused enough damage to the windshields of spacecraft that the
windows had to be replaced (Space Debris and Human Spacecraft). In addition to damaged
windows, images show that even objects less than half an inch in diameter can puncture through the
structure of the space shuttle Endeavour (Trouble in orbit: the growing problem of space junk). The
fact that such puny pieces of junk can cause such severe damage upon space shuttles suggests that
spacecraft are mere inflated balloons to space due to the high velocities objects are travelling. This
only makes situations worse for humans as there is no really protected inside or

Project 1 Essay
Project 1:
1. Results for a: ht=500/ spd=0mph –too slow! Satellite crashed into Earth! Results for b: ht=500/
spd=100mph –too slow! Satellite crashed into Earth! Results for c: ht=500/ spd=25000mph –too
fast! Satellite left viewing area and took off in space somewhere!
I was determined to get a 100%, so I played around with the numbers until I found that at 500 miles
above the Earth, the satellite would have to be launched at 16,700mph in order to reach a perfect
orbit. Below is the explanation of why:
"Although the satellite is always falling towards the earth, because of gravity, the right combination
of height and speed causes the satellite to fall as much as the earth curves, so the satellite never
crashes"
2. I thought ... Show more content on Helpwriting.net ...
7. The asteroid that is travelling at constant speed would not be accelerating, since constant speed is
continuous motion with no acceleration.
8. Gravity is the force that is keeping the asteroid in orbit around the sun. The force of gravity will
pull the asteroid inward, but because the asteroid is constantly moving forward– it is sort of a
constant state of perfectly balanced tug–of–war.
9. 1) Sun 2)Jupiter 3) Mars 4)Other Asteroids The reason for these rankings is because although the
Asteroid belt is located between Jupiter and Mars, ALL of the other objects in these ranks are
orbiting around the Sun... Jupiter's gravitational pull will cause an asteroid to collide with another
asteroid, however, so you could say that Jupiter's influence is stronger than Mars, and if other
asteroids collide with an asteroid, it will change the mass of the asteroid itself by breaking it apart–
only to be pulled in again by Mars or Jupiter (and overall, the Sun).
10. KE= 0.5 x 1000 (30)2 (that is 30 squared..I just cant figure out how to get my keyboard to type it
and it's driving me insane...)
11. KE= 0.5x1000(30)squared KE= 0.5x1000(900) KE= 0.5x900,000 KE= 450,000

12. If the distance of the sun were farther, the kinetic energy would not change– because the velocity
and the mass of asteroid would still remain the

Taking a Look at Orbital Mechanics
Orbital mechanics is the application of ballistic and celestial mechanics to motion, especially
pertaining to rockets and spacecraft. Many famous physicists and mathematicians have helped
develop equations, formulas, and laws to understand different aspects of orbital mechanics; such as
Newton with centripetal force and the gravitational constant (GM) and also Kepler and his three
laws of planetary motion. Though my interest did not sprout from who was involved with orbital
mechanics, but the orbital mechanics themselves. I've always had an interest in astronomy and
physics, and this seems like the perfect combinations of both. There are six general aspects of orbital
mechanic elements eccentricity (e), semi–major axis (a), inclination (i), argument of periapsis (ω),
time of periapsis passage (T), and longitude of ascending node (Ω).
[I will help define these with use of a diagram put I do not have the diagram drawn this is also the
diagram of observations of an object in orbit talked about in the outline]
Though these are not the only important elements of orbital mechanics, there is also the period (P)
and true anomaly (v) which is "the angular distance of a point in an orbit past the point of periapsis.
Conic sections and conics are use in orbital mechanics to determine the type of orbit and defining
the eccentricity of the conic section and the hyperbola or parabola formed within the right cone
depending on the angularity of the section. All

Math Of Cheerleading Essay
The Math of Stunting
The physical nature of the sport cheerleading, demands a grasp on the knowledge of how both the
forces that are exerted and those one experiences will affect one's performance in a stunt or trick.
Physics and mathematics can be found everywhere in cheerleading: 8–counts direct a dancing
routine, weight distribution determines a three–level pyramid. The highly organized nature of
cheerleading, with all of its formations and different groupings of, in Columbia River's case,
members based on their "job" in that stunt – back spot, base, or flyer – can hardly be removed from
the concepts of math. Rather, a cheerleader with a true knowledge of physics and mathematics will
undoubtedly be more successful in their happenings – ... Show more content on Helpwriting.net ...
If we measured in meters then a=4.9.) t is the time in seconds, v0 is the initial velocity of 30 feet per
second, and s0 is the initial height or 4 feet. Thus we have: h(t)=–16t^2+30t+4 as a model for the
height.
Note that we disregard complicating factors such as air resistance, spin, because the uncertainty and
unpredictability of these factors leads to a much more complicated equation than needed.
(a) Will the flyer reach 20 feet?
Let h(t)=20 and solve:
20=–16t^2+30t+4
–16t^2+30t–16=0
Using the quadratic formula, x=(–b±√(b^2–4ac))/2a , we can solve for t: x=(–30±√(〖30〗^2–4(–
16)(–16)))/(2(–16)) The discriminant, x=–b±√(b^2–4ac) , is –124 so there are no real solutions. The
flyer will never reach 20 feet with the given initial velocity.
The graph of height vs time:
Height (in feet)
Time (in seconds) (b) What initial velocity is required to achieve a height of 25 ft.?
Let h(t)=25 and solve for V_0: 25=–16t^2+v_0 t+4 –16t^2+v_0 t=21 v_0 t=21+16t^2 v_0=21/t+16t
We want to minimize this function. Using calculus we find the minimum to be at t approximately
1.15 seconds and velocity approximately 36.66 feet per

Kepler 's Laws Of Planetary Motion
Kepler's Laws Johannes Kepler formulated the Three Laws of Planetary Motion. The first is the Law
of Orbits, stating that all planets move in elliptical orbits with the sun at one focus. The second law
is the Law of Areas. This is the idea that a line that connects the planet to the sun sweeps out equal
areas in the plane of the planet's orbit in equal time intervals. Last is the Law of Periods which states
that the square of the period of any planet is proportional to the cube of the semi major axis of its
orbit. It is simple to just accept the laws and memorize them, but take it a step further. What are the
explanations to these laws and how did Kepler come to those conclusions? The results that Kepler
published actually began with Tycho Brahe. Brahe was interested in reforming astronomy, and
wanted to prove his idea that planets circle the sun which then move around the stationary earth. He
made several observations with advanced instruments that still to this day have been proven by
calculations done with modern computers to be quite accurate. All of Brahe's observations were very
useful and interesting to Kepler who became Brahe's assistant in 1600. Some propose that Brahe
only continued making observations because he was unsure of what to do with them or how to apply
them, so it was left up to assistants such as Johannes Kepler to put the information to use (J.V.
Field). Kepler had been engaging in finding an orbit of Mars, and Brahe's observations

Johannes Kepler's Research Paper
Kepler's Discoveries "I much prefer the sharpest criticism of a single intelligent man to the
thoughtless approval of the masses." Said Johannes Kepler in deep thought. He developed smallpox
when he was three, which left him partially disabled and with poor vision. Even though he was
disabled he still was a straight A student who scored well in math. His dream was to enter the
ministry, but the economy forced him to pursue mathematics, which later he realized was Gods
leading.
The First Law of Planetary Motion, which is The Law of Ellipses, states that The path of the planets
around the sun is elliptical in shape, with the center of the sun being located at one focus. It took
five years of measuring the motion of Mars across the sky for

Persuasive Essay On Space Travel
No one wants to die. That is a fact of life and apart of human nature. Unfortunately, humans are the
main source for increasing pollution rates, deforestation, a large contributor for climate change, and
the overall destruction of our planet. This is not a local issue, a state issue, or even an issue
subjected to just one country. This is a global issue that will effect everything. Our withering planet
is why space travel is important for our society.
Earth is our home. In fact, it's home to over 7.5 billion people. All of those people using gas, eating
food, producing waste: biological and material, in need of clothes, and reproducing. With current
trends in population growth, many analyst speculate that the world population will expand to 10
billion people in 2050. Humans typically cannot fathom numbers larger than the thousands, as a
local point of reference, a very obnoxiously crowded Cowboy's Stadium can hold a maximum of
100,000 people, now imagine that number times 100,000. David Criswell, a physicist from the
University of Houston, claims "We are already well beyond what the biosphere can provide. We
have to go outside to get something else". Earth has reached its threshold. It can barely support 7.5
billion people, we have a pollution crisis we are already trying to deal with, another 2.5 billion
people will probably cause us to suffocate in our own man–made waste. NASA sums it up perfectly,
the only logical thing left to do is establish a space colony.
The issue

Vector
VECTOR FUNCTIONS
VECTOR FUNCTIONS
Motion in Space: Velocity and Acceleration
In this section, we will learn about:
The motion of an object using tangent and normal vectors.
MOTION IN SPACE: VELOCITY AND ACCELERATION
Here, we show how the ideas of tangent and normal vectors and curvature can be
used in physics to study:
 The motion of an object, including its velocity and acceleration, along a space curve.
VELOCITY AND ACCELERATION
In particular, we follow in the footsteps of
Newton by using these methods to derive
Kepler's First Law of planetary motion.
VELOCITY
Suppose a particle moves through space so that its position vector at
time t is r(t).
VELOCITY
Vector 1
Notice from the figure ... Show more content on Helpwriting.net ...
 So, C=i–j+ k

VELOCITY & ACCELERATION
Example 3
It follows: v(t) = 2t2 i + 3t2 j + t k + i – j + k
= (2t2 + 1) i + (3t2 – 1) j + (t + 1) k
Example 3
Since v(t) = r'(t), we have: r(t) = ∫ v(t) dt
= ∫ [(2t2 + 1) i + (3t2 – 1) j + (t + 1) k] dt = (⅔t3 + t) i + (t3 – t) j + (½t2 + t) k + D
Example 3
Putting t = 0, we find that D = r(0) = i.
So, the position at time t is given by: r(t) = (⅔t3 + t + 1) i + (t3 – t) j + (½t2 + t) k
The expression for r(t) that we obtained
in Example 3 was used to plot the path of the particle here for 0 ≤ t ≤ 3.
In general, vector integrals allow us to recover:
 Velocity, when acceleration is known
v(t )
v(t0 )
t t0
a(u ) du
 Position, when velocity is known
r (t ) r (t0 )

t t0
v(u ) du
If the force that acts on a particle is known, then the acceleration can be found from
Newton's Second Law of Motion.
The vector version of this law states that if,
at any time t, a force F(t) acts on an object of mass m producing an acceleration a(t),
then
F(t) = ma(t)
Example 4
An object with mass m that moves in
a circular path with

R1 Ariane Research Paper
R1– Ariane is a disposable rocket, that carries charges for low altitude orbits (300km to 2000km
altitude), it can carries from 6 to 9,5 tons, depending on the orbit of the charge (lower more, higher
less), the company responsible for the production is the EADS SPACE transportation, this company
is supervisioned by ESA, and the rocket is operated by the company Arianespace. The rocket Ariane
5, on June of 1996, exploded 40 seconds after launch, on an altitude of 3.7 km, the debris of the
rocket felt in an area of 5x2.5 km^2, in an area of hard access, because of it, not all the parts of the
rocket were found, the sequence that originated the explosion was: – 22 seconds after the hour zero
(H0) there was a variation of 10Hz in the hydraulic pressure in ... Show more content on
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Later in the analysis of the data about the accident, is was found that the same system used in the
rocket Ariane 5, was used in the rocket Ariane 4, the cause for the system failure was an error in a
subroutine in the computer responsible for the Inertial Reference (SRI), the error was an "operand
error", which happened when a 64–bit floating point couldn't be converted to 16–bit signed integer,
the value was too large to be converted, and caused the software exception, and the code wasn't
ready to handle this type of error (exception handling). The software used in the Ariane 4 was
clearly not suitable for the Ariane 5, the variation of behavior between the two launchings was too
big, the alignment function was operative based in the data of the Ariane 4, where the system
operable for the first 40 seconds of flight could handle the data (based on acceleration), this
characteristics was valuable only for this rocket, but not usable in Ariane 5, due this rocket had
higher initial acceleration (5x more), it made the system stop working (OperandError), and resulted
in the chain of occurrences, causing malfunction and subsequently the

Universal Gravitation Essay
Newton's law of universal gravitation extends gravity beyond earth. The law of the universal
gravitation is about the universality of gravity. Newton's law of universal gravitation states that
every particle is attached with each other particle in the universe. Newton's law of the universal
gravitation states that, "Any two bodies in the universe attract each other with a force that is directly
proportional to the product of their masses and inversely proportional to the square of the distance
between them." The law of universal gravitation was find by Newton because of an apple. When the
apple fell from the tree on to Newton's head. Newton found it mysteriously, so he started to think
about the reason and the formula out of this nature phenomenon. ... Show more content on
Helpwriting.net ...
We as humans live in the Earth. Earth is one of the planets, and it is also stays in the orbit. It stays
on the orbit and circled round the sun. The days it takes for one circle of the Earth's orbit is one year
and move circled around. Our daily life can be break down as day and night. The difference between
it is the sun. During the day, we have sunshine but not at the night. It is a rotate of the daily life. It
will never change, there will not be one day is without sunshine. In some ways, the universal
gravitation, and the principles of the planets stay in the orbit around the sun affects our society.
Newton's law of universal gravitation extends gravity beyond earth. The law of the universal
gravitation is about the universality of gravity. Newton's law of universal gravitation states that
every particle is attached with each other particle in the universe. Newton's law of the universal
gravitation states that, "Any two bodies in the universe attract each other with a force that is directly
proportional to the product of their masses and inversely proportional to the square of the distance
between

The Life of Johannes Kepler Essay
The Life of Johannes Kepler
HIS LIFE
Johannes Kepler was a German astronomer and mathematician ho discovered that planetary motion
is elliptical. Early in his life, Kepler wanted to prove that the universe obeyed Platonistic
mathematical relationships, such as the planetary orbits were circular and at distances from the sun
proportional to the Platonic solids (see paragraph below). However, when his friend the astronomer
Tycho Brahe died, he gave Kepler his immense collection of astronomical observations. After years
of studying these observations, Kepler realized that his previous thought about planetary motion
were wrong, and he came up with his three laws of planetary motion. Unfortunately, he did not have
a unifying theory for ... Show more content on Helpwriting.net ...
Each interior angle of an equilateral triangle is 60°, therefore we could fit together three, four, or
five of them at a vertex, and these correspond to the tetrahedron, the octahedron, and the
icosahedron. Each interior angle of a square is 90°, so we can fit only three of them together at each
vertex, giving us a cube. The interior angles of the regular pentagon are 108°, so again we can fit
only three together at a vertex, giving us the dodecahedron.
That makes five regular polyhedra. However, what would happen if we had a six–sided figure?
Well, its interior angles are 120°, so if we fit three of them together at a vertex the angles add up to
360°, and therefore they lie flat. For this reason we cannot use hexagons to make a Platonic solid. In
addition, obviously, no polygon with more than six sides can be used either, because the interior
angles just keep getting larger.
The Greeks, who had to find religious truth in mathematics, found the idea of exactly five Platonic
solids very compelling. The philosopher Plato concluded that they must be the fundamental building
blocks of nature, and assigned to them what he believed to be the essential elements of the universe.
He followed the earlier philosopher Empedocles in assigning fire to the tetrahedron, earth to the
cube, air to the octahedron, and water to the icosahedron. To the dodecahedron, Plato assigned the

Kepler's Laws Essay examples
Kepler's Laws
In today's world, we have very advanced technology. There have been many new technological and
medical advancements as we entered the new century. The Internet allows us to shop, talk, and find
valuable information on very scarce topics, and even check stocks with a simple click of a button.
Medical advancements had recently been discovered on "The Human Genome Projects," the first
gene was mapped and within a short period of time we will have mapped out all the genes in a
human chromosome. This is absolutely amazing because we will now be able to reveal the many
causes of serious deadly diseases. Throughout the years, we have gained the technology to send
astronauts into space to gather new information about our ... Show more content on Helpwriting.net
...
Brahe did not trust Kepler and worried that Kepler would surpass him and become more well known
than him. So Brahe assigned him the job of understanding the orbit of Mars because it gave Brahe
much difficulty, and would keep Kepler occupied while Brahe worked on his theory of the solar
system. Brahe also kept other data from Kepler hoping that he would be the one to discover the
orbits of the planets before Kepler did. In 1601, when Brahe died Kepler received all of Brahe's
data. Whether he obtained this data legally or not is still in debate today, however, it is fortunate that
he obtained this data. (Silverberg, 160)
When Kepler first began his work on the orbit of Mars he was under the assumption, as many
scientists were, that the planetary orbits were circular, and that the Sun was at the center of the
orbits. This type of system is called a heliocentric system. Also at this time only six planets were
known. When Kepler obtained Brahe's data he discovered that the orbits were not perfect circles, but
instead were ellipses that were only slightly flattened. The reason nobody else realized this was
because the orbits were so slightly elliptical that extensive investigation and data would be needed
to show this. It also turned out that the reason the orbit of Mars was very difficult to understand was
because its orbit was more eccentric than the other planets that Kepler and Brahe had data about. To
understand a lot of Kepler's work you must first

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