The First Satellites

8/3/2019 The First Satellites

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KENNETH T. GUILLERMO FN-RESEARCHWORKBSGE 05 SATELLITE GEODESY

THE FIRST SATELLITES

In the mid-1950s, the desire to learn more about space led to thedevelopment of satellites. A satellite is a man-made object put intoorbit around a larger body. Scientists, astronomers, and others wereeager to learn about the new space environment, and hopefully makebreakthroughs in communications, weather, and manned space flight.

First International Satellite Plans

The United States and Russia were the first two nations to drawplans for an artificial satellite. In 1955, the United States andRussia publicly announced their proposals concerning the constructionof satellites. Within two years, Russia accomplished their goal, and

the United States followed closely behind.

The first artificial satellite was Sputnik 1, launched by theSoviet Union on October 4, 1957, and initiating the SovietSputnikprogram, with Sergei Korolevas chief designer (there is a crater onthe lunar far side which bears his name). This in turn triggeredthe Space Race between the Soviet Union and the United States.

Sputnik 1 helped to identify the density of high atmosphericlayers through measurement of its orbital change and provided data onradio-signal distribution in the ionosphere. The unanticipatedannouncement of Sputnik 1's success precipitated the Sputnik crisis inthe United States and ignited the so-called Space Race within the Cold

War.Sputnik 2 was launched on November 3, 1957 and carried the first

living passenger into orbit, a dog named Laika.

In May, 1946, Project RAND had released the Preliminary Design ofan Experimental World-Circling Spaceship, which stated, "A satellitevehicle with appropriate instrumentation can be expected to be one ofthe most potent scientific tools of the Twentieth Century. The UnitedStates had been considering launching orbital satellites since 1945under the Bureau of Aeronautics of the United States Navy. The UnitedStates Air Force's Project RAND eventually released the above report,but did not believe that the satellite was a potential militaryweapon; rather, they considered it to be a tool for science, politics,

and propaganda. In 1954, the Secretary of Defense stated, "I know ofno American satellite program."

On July 29, 1955, the White House announced that the U.S.intended to launch satellites by the spring of 1958. This became knownas Project Vanguard. On July 31, the Soviets announced that theyintended to launch a satellite by the fall of 1957.

Following pressure by the American Rocket Society, the NationalScience Foundation, and the International Geophysical Year, military
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interest picked up and in early 1955 the Army and Navy were workingon Project Orbiter, two competing programs, the army's which involvedusing a Jupiter C rocket, and the civilian/Navy Vanguard Rocket, tolaunch a satellite. At first, they failed: initial preference wasgiven to the Vanguard program whose launch vehicle had a strange anduncanny way of exploding on national television. But finally, three

months after Sputnik 1, the project succeeded;Explorer 1 thus becamethe United States' first artificial satellite on January 31, 1958.

In June 1961, three-and-a-half years after the launch of Sputnik1, the Air Force used resources of the United States SpaceSurveillance Network to catalog 115 Earth-orbiting satellites.

The largest artificial satellite currently orbiting the Earth isthe International Space Station.

Sputnik 1

Sputnik 1 was the first artificial satellite to bring the Space

Age to life. On October 4, 1957, the Soviet Union launched Sputnik 1.Its official Russian name was Iskustvennyi Sputnik Zemli, or "FellowTraveler of the Earth." Sputnik 1 was launched by Russia's Old NumberSeven rocket at Baikonur Cosmodrome. The once secret cosmodrome iswhat makes Russia lead the rest of the world in launching men andmachines into space month after month. Sputnik 1 was described as "asilver-zinc battery and a radio transmitter in a 23 inch polishedaluminum ball" (Curtis 1). The satellite was also pressurized withnitrogen circulated by a cooling fan. Two eight-feet and two ten-feetradio antenna whips were secured to the outside of the satellite totransmit radio signals. "For three weeks, as it twirled around theworld every 96 minutes in a globe-girdling orbit 588 miles above our

heads, Sputnik beep-beeped its visionary message of a future above theocean of air" (Curtis 1). After 92 days, Sputnik 1 burned as it fellfrom orbit into the atmosphere January 4, 1958.

Sputnik 1: The first artificial satellite to orbit Earth.
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Explorer 1

Explorer 1 was the first satellite launched by the United states.On January 31, 1958 the United States of America's Jupiter-C rocketlaunched Explorer 1 at Cape Canaveral. The Army was responsible forthe preparation of Explorer 1. The army was asked by Washington

officials to try to send a satellite to orbit because they wereworried about losing prestige. Four months after Russia orbitedSputnik 1 the United States entered the space race as well.

International Satellite Builders

Many other nations in Asia and Europe soon joined the race inspace by launching satellites. "The majority of satellites have beenbuilt by Russia and the United States, but the countries of WesternEurope in the European Space Agency, Japan, China, India, Canada,Israel, Brazil and others are actively engaged in satellite

development" (Curtis 6). France's Diamont rocket launched its firstsatellite Asterix 1 in Algeria on November 26, 1965. On February 11,1970 Japan's Lambda 4S-5 rocket launched its first satellite Ohsumifrom Kagoshima. China's Long March-1 rocket soon followed launchingits first satellite Mao 1 from Inner Mongolia on April 24, 1970. Ayear and a half later on October 28, Britain's Black Arrow rocketlaunched its first satellite Black Knight 1 from Woomera Australia.Europe's rocket Ariane launched its first satellite CAT from Kourou inFrench Guiana on December 24, 1979. Rohini 1, the first satellite madeby India, was launched from Sriharikota Island on July 18,1980.Israel_s Shavit rocket fired its first satellite Horizon 1 from NegevDesert on September 19,1988. Iraq followed a year later when it

launched Rocket 3rd Stage from Al-Anbar on December 5.

Conclusion

The first satellites led the way to most of our knowledgeconcerning space today. Because of the success of many of the firstsatellites, extensive research could be done about the Solar Systemusing the pictures and information retrieved by the satellites. Since1957, more than 4100 satellites have successfully been launched. Withall the technology created day after day, our knowledge of space hasbecome very sophisticated and will continue to grow.


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GRAVITY FIELD OF THE EARTH

Gravity is a force exerted by an object of mass. The gravitycreated by the Earths mass creates a field around the Earth. Thegravity field of the Earth is an equipotential surface. Theequipotential surface, or normal gravity, is the perpendicular of the

Earths gravitational force and the centrifugal force caused by therotation of the Earth. The equipotential surface is called the geoid.For general purposes, the mean sea level is assumed as the surface ofthe geoid.

The Earths gravity field is dynamic, or always changing. Thegravity field is dynamic, because the Earth is a living planet. TheEarths mass is constantly shifting and being redistributed. Theshifting and redistribution mainly comes from volcanoes (lava fromEarths mantle), tectonic plate movement, and ocean tides.

Observed gravity values often differ from normal gravity values.

The difference is caused by the topography of the Earth, the Earthsrotation (flattening of the poles), and the dynamic features of theEarth. A gravity anomaly is the difference between the observedreduced gravity and the normal gravity. An observed reduced gravityvalue comes from reducing the observed gravity value to the geoid.Various reduction methods can accomplish this, and each has their ownunique method of reduction. For instance, the free air correctionremoves the topography above the geoid, while a Bouguer reductionaccounts for the masses above the geoid. Each type of reductionproduces a type of gravity anomaly. Additional anomalies, in theobserved gravity, may exist; these are caused by certain types ofdense rocks, rock structures, and other unknown factors.

There are two ways to measure gravity: absolute measurement andrelative measurement. Absolute measurement measures the time it takesa falling object to go a known distance. An instrument that is usedto make absolute gravity measurements is called an absolutegravimeter. Modern absolute gravimeters use retro reflectors andlasers. Since an absolute gravimeter measures gravity at one specificlocation, an interferometer is used to combine multiple absolutegravimeters in order to measure the gravity gradient.

Relative measurement measures the difference in gravity betweentwo points. An instrument that is used to make relative gravity

measurements is called a relative gravimeter. Relative gravimetersuse an attached spring with a weighted object hanging at one end; alsoknown as a mass-spring system. The amount the spring stretches isproportional to the gravitational force. The units of gravity areGal, or 1 cm / sec.

According to Thapa (2007), a modern absolute gravimeter is accurate upto 1 Gal(mircoGal), and a modern relative gravimeter is accurate upto 0.01 mGal (milliGals).


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GRAVITATION AND GRAVITY FIELD OF THE EARTH

The gravity of Earth, denoted g, refers to the acceleration thatthe Earth imparts to objects on or near its surface.

In SI units this acceleration is measured in metres per second persecond (in symbols, m/s2 or ms-2) or equivalently in newtons perkilogram (N/kg or Nkg-1). It has an approximate value of 9.81 m/s2,which means that, if we ignore the effects of air resistance (whichare often negligible), the speed of an object falling freelynear theEarth's surface will increase by about 9.81 metres (about 32ft) persecond every second. Although the precise strength varies depending onlocation, the nominal "average" value at the Earth's surface, knownas standard gravity is, by definition, 9.80665 m/s2 (32.1740 ft/s2).

This quantity represents the strength of the Earth's gravity andis often referred to informally as little g. (In contrast,the gravitational constant G is similarly referred to as big G.) This

quantity is denoted variously as gn, ge (though this sometimes meansthe normal equatorial value on Earth, 9.78033 m/s2), g0, gee, orsimply g (which is also used for the variable local value). Thesymbol g should not be confused with g, the abbreviation for gram(which is not italicized).[1][2]

There is a direct relationship between gravitationalacceleration and the downwards weight force experienced by objects onEarth (see Conversion between weight and mass). However, other factorssuch as the rotation of the Earth also contribute to the netacceleration.

VARIATION IN GRAVITY AND APPARENT GRAVITYEarth's gravitational force is often modeled as though the Earth

were an inertsphere of uniform density. Such a body would produce afield of uniform magnitude and direction at all points on its surface.In reality, there are slight deviations in both the magnitude anddirection of gravity across the surface of the Earth because none ofthose qualities are exactly true of Earth.

Furthermore, the net force exerted on an object due to the Earth,called apparent gravity or effective gravity varies due to thepresence of other forces. A scale or plumb bob measures only thiseffective gravity.

The strength of Earth's apparent gravity varieswith latitude, altitude, local topography and geology.

LATITUDE

At latitudes nearer the equator, the outward centrifugalforce produced by Earth's rotation is stronger than at polarlatitudes. This counteracts the Earth's gravity to a small degree,
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reducing downward acceleration of falling objects. At the equator,this apparent gravity is 0.3% less than actual gravity.

Gravity provides centripetal force, keeping objects on thesurface (and indeed the surface itself) moving in a circular motion.Consider that if the gravity of the Earth were to shut off, objectswould fly off into space in the direction of their motion inaccordance with Newton's First Law of Motion. Alternatively, ifEarth's gravity were weakened so as to match the centrifugal force(at, say, the equator where rotational speed is largest) then objectsthere would appear to float. Also, at the poles, only gravity wouldcontribute to weight and objects would not float. In this sense, localgravity (gravity at a particular point on the surface of the Earth)felt as weight is gravity due to the Earth's mass minus thecentrifugal force.

The second major cause for the difference in gravity at differentlatitudes is that the Earth's equatorial bulge (itself also caused bycentrifugal force) causes objects at the equator to be farther from

the planet's centre than objects at the poles. Because the force dueto gravitational attraction between two bodies (the Earth and theobject being weighed) varies inversely with the square of the distancebetween them, an object at the equator experiences a weakergravitational pull than an object at the poles.

In combination, the equatorial bulge and the effects ofcentrifugal force mean that sea-level gravitational accelerationincreases from about 9.780 ms2 at the equator to about 9.832 ms2 atthe poles, so an object will weigh about 0.5% more at the poles thanat the equator.

The same two factors influence the direction of the effectivegravity. Anywhere on Earth away from the equator or poles, effective

gravity points not exactly toward the centre of the Earth, but ratherperpendicular to the surface of the geoid, which, due to the flattenedshape of the Earth, is somewhat toward the opposite pole. About halfof the deflection is due to centrifugal force, and half because theextra mass around the equator causes a change in the direction of thetrue gravitational force relative to what it would be on a sphericalEarth.

ALTITUDE

Gravity decreases with altitude, since greater altitude meansgreater distance from the Earth's centre. All other things being

equal, an increase in altitude from sea level to the top of MountEverest (8,850 metres) causes a weight decrease of about 0.28%. (Anadditional factor affecting apparent weight is the decrease in airdensity at altitude, which lessens an object's buoyancy.) It is acommon misconception that astronauts in orbit are weightless becausethey have flown high enough to "escape" the Earth's gravity. In fact,at an altitude of 400 kilometres (250 miles), equivalent to a typicalorbit of the Space Shuttle, gravity is still nearly 90% as strong as
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at the Earth's surface, and weightlessness actually occurs becauseorbiting objects are in free-fall.[6]

The following formula approximates the Earth's gravity variation withaltitude:

Where

is the gravity measure at height above sea level. is the Earth's mean radius. is the standard gravity.

Depth

If the Earth were a sphere of uniform density then gravity woulddecrease linearly to zero as one travelled in a straight line from the

Earth's surface to its centre. This is a consequence of Gauss' law forgravity. Because of the spherical symmetry, gravity is radiallydownward and equal in magnitude at all points at a given radius r. Thesurface area of a sphere of radius r being 4r2, Gauss's law gives

where G is the gravitational constant and M is the total mass enclosedwithin the surface. Since, for r less than the Earth's radius and aconstant density , M = (4/3)r3, the dependence of gravity on depthis

If the density decreases linearly with depth from a density 0 atthe center to 1at the surface, then (r) = 0 - (0-1)r / re, and

MATHEMATICAL MODELS

If the terrain is at sea level, we can estimate g:

where= acceleration in ms2 at latitude :

This is the International Gravity Formula 1967, the 1967 GeodeticReference System Formula, Helmert's equation or Clairaut's formula.

The first correction to this formula is the free air correction(FAC), which accounts for heights above sea level. Gravity decreaseswith height, at a rate which near the surface of the Earth is such
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that linear extrapolation would give zero gravity at a height of onehalf the radius is 9.8 ms2 per 3 200 km.

Using the mass and radius of the Earth:

The FAC correction factor (g) can be derived from the definition

of the acceleration due to gravity in terms of G, the GravitationalConstant (see Estimating g from the law of universal gravitation,below):

where:

At a height h above the nominal surface of the earth gh is given

by:

So the FAC for a height h above the nominal earth radius can beexpressed:

Collecting terms, simplifying and neglecting small terms(h


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cancel at a surface rock density of 4/3 times the average density ofthe whole Earth.)

For the gravity below the surface we have to apply the free-aircorrection as well as a double Bouguer correction. With the infiniteslab model this is because moving the point of observation below theslab changes the gravity due to it to its opposite. Alternatively, wecan consider a spherically symmetrical Earth and subtract from themass of the Earth that of the shell outside the point of observation,because that does not cause gravity inside. This gives the sameresult.

Helmert's equation may be written equivalently to the versionabove as either:

or

An alternate formula for g as a function of latitude is the WGS(World Geodetic System) 84 Ellipsoidal Gravity Formula:

The difference between the WGS-84 formula and Helmert's equationis less than 0.68 106 ms2.

From the law of universal gravitation, the force on a body acted

upon by Earth's gravity is given by

where r is the distance between the centre of the Earth and the body,

and here we take m1 to be the mass of the Earth and m2 to be the mass

of the body.

Additionally, Newton's second law, F = ma, where m is mass

and a is acceleration, here tells us that

Comparing the two formulas it is seen that:
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So, to find the acceleration due to gravity at sea level,

substitute the values of the gravitational constant, G, the

Earth's mass (in kilograms), m1, and the Earth's radius (in metres), r,

to obtain the value of g:

Note that this formula only works because of the mathematical

fact that the gravity of a uniform spherical body, as measured on or

above its surface, is the same as if all its mass were concentrated at

a point at its centre. This is what allows us to use the Earth's

radius for r.

The value obtained agrees approximately with the measured value

of g. The difference may be attributed to several factors, mentioned

above under "Variations":

The Earth is not homogeneous The Earth is not a perfect sphere, and an average value must be

used for its radius

This calculated value of g only includes true gravity. It doesnot include the reduction of constraint force that we perceive as

a reduction of gravity due to the rotation of Earth, and some ofgravity being "used up" in providing the centripetal acceleration

There are significant uncertainties in the values of r and m1 as

used in this calculation, and the value of G is also rather difficult

to measure precisely.

If G, g and r are known then a reverse calculation will give an

estimate of the mass of the Earth. This method was used by Henry

Cavendish.
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Escape velocity is actually a speed (not a velocity) because itdoes not specify a direction: no matter what the direction of travelis, the object can escape the gravitational field (though its path mayintersect the planet). The simplest way of deriving the formula forescape velocity is to use conservation of energy. Imagine that aspaceship of mass m is at a distance r from the center of mass of the

planet, whose mass is M. Its initial speed is equal to its escapevelocity,ve. At its final state, it will be an infinite distance awayfrom the planet, and its speed will be negligibly small and assumed tobe 0. Kinetic energy K and gravitational potential energy Ug are theonly types of energy that we will deal with, so by the conservation ofenergy,

K = 0 because final velocity is zero, and Ug = 0 because its finaldistance is infinity, so

Defined a little more formally, "escape velocity" is the initialspeed required to go from an initial point in a gravitationalpotential field to infinity with a residual velocity of zero, with allspeeds and velocities measured with respect to the field.Additionally, the escape velocity at a point in space is equal to thespeed that an object would have if it started at rest from an infinitedistance and was pulled by gravity to that point. In common usage, the

initial point is on the surface of a planet or moon. On the surface ofthe Earth, the escape velocity is about 11.2 kilometers persecond (~6.96 mi/s), which is approximately 34 times the speed ofsound (Mach 34) and several times the muzzle velocity of a riflebullet (up to 1.7 km/s). However, at 9,000 km altitude in "space", itis slightly less than 7.1 km/s.

The escape velocity relative to the surface of a rotating bodydepends on direction in which the escaping body travels. For example,as the Earth's rotational velocity is 465 m/s at the equator, a rocketlaunched tangentially from the Earth's equator to the east requires aninitial velocity of about 10.735 km/s relative to Earth to escapewhereas a rocket launched tangentially from the Earth's equator to the

west requires an initial velocity of about 11.665 km/s relative toEarth. The surface velocity decreases with the cosine of thegeographic latitude, so space launch facilities are often located asclose to the equator as feasible, e.g. the American CapeCanaveral (latitude 2828' N) and the French Guiana SpaceCentre (latitude 514' N).

The barycentric escape velocity is independent of the mass of theescaping object. It does not matter if the mass is 1 kg or 1,000 kg,what differs is the amount of energy required. For an object of
http://en.wikipedia.org/wiki/Planethttp://en.wikipedia.org/wiki/Natural_satellitehttp://en.wikipedia.org/wiki/Km/shttp://en.wikipedia.org/wiki/Km/shttp://en.wikipedia.org/wiki/Muzzle_velocityhttp://en.wikipedia.org/wiki/Trigonometric_functionhttp://en.wikipedia.org/wiki/Cape_Canaveral_Air_Force_Stationhttp://en.wikipedia.org/wiki/Cape_Canaveral_Air_Force_Stationhttp://en.wikipedia.org/wiki/Guiana_Space_Centrehttp://en.wikipedia.org/wiki/Guiana_Space_Centrehttp://en.wikipedia.org/wiki/Guiana_Space_Centrehttp://en.wikipedia.org/wiki/Guiana_Space_Centrehttp://en.wikipedia.org/wiki/Cape_Canaveral_Air_Force_Stationhttp://en.wikipedia.org/wiki/Cape_Canaveral_Air_Force_Stationhttp://en.wikipedia.org/wiki/Trigonometric_functionhttp://en.wikipedia.org/wiki/Muzzle_velocityhttp://en.wikipedia.org/wiki/Km/shttp://en.wikipedia.org/wiki/Km/shttp://en.wikipedia.org/wiki/Natural_satellitehttp://en.wikipedia.org/wiki/Planet


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mass m the energy required to escape the Earth's gravitational fieldis GMm / r, a function of the object's mass (where r is the radius ofthe Earth, G is the gravitational constant, and M is the mass ofthe Earth, M=5.97361024 kg).

To expand upon the derivation given ,

where ve is the barycentric escape velocity, G is the gravitationalconstant, M is the mass of the body being escaped from, r isthe distance between the center of the body and the point at whichescape velocity is being calculated, g is the gravitationalaccelerationat that distance, and is the standard gravitationalparameter.

The escape velocity at a given height is times the speed in acircular orbit at the same height, (compare this with equation (14)

in circular motion). This corresponds to the fact that the potentialenergy with respect to infinity of an object in such an orbit is minustwo times its kinetic energy, while to escape the sum of potential andkinetic energy needs to be at least zero. The velocity correspondingto the circular orbit is sometimes called the first cosmic velocity,whereas in this context the escape velocity is referred to asthe second cosmic velocity"[5]

For a body with a spherically-symmetric distribution of mass, thebarycentric escape velocity ve from the surface (in m/s) isapproximately 2.364105 m1.5kg0.5s1 times the radius r (in meters)times the square root of the average density (in kg/m), or:

The barycentric gravitational acceleration can be obtained fromthe gravitational constant G and the mass of Earth M:

where r is the radius of Earth. Thus

so the two derivations given above are consistent.
http://en.wikipedia.org/wiki/Gravitational_constanthttp://en.wikipedia.org/wiki/Earthhttp://en.wikipedia.org/wiki/Gravitational_constanthttp://en.wikipedia.org/wiki/Gravitational_constanthttp://en.wikipedia.org/wiki/Masshttp://en.wikipedia.org/wiki/Distancehttp://en.wikipedia.org/wiki/Gravitational_accelerationhttp://en.wikipedia.org/wiki/Gravitational_accelerationhttp://en.wikipedia.org/wiki/Standard_gravitational_parameterhttp://en.wikipedia.org/wiki/Standard_gravitational_parameterhttp://en.wikipedia.org/wiki/Circular_motionhttp://en.wikipedia.org/wiki/Escape_velocity#cite_note-5http://en.wikipedia.org/wiki/Escape_velocity#cite_note-5http://en.wikipedia.org/wiki/Circular_motionhttp://en.wikipedia.org/wiki/Standard_gravitational_parameterhttp://en.wikipedia.org/wiki/Standard_gravitational_parameterhttp://en.wikipedia.org/wiki/Gravitational_accelerationhttp://en.wikipedia.org/wiki/Gravitational_accelerationhttp://en.wikipedia.org/wiki/Distancehttp://en.wikipedia.org/wiki/Masshttp://en.wikipedia.org/wiki/Gravitational_constanthttp://en.wikipedia.org/wiki/Gravitational_constanthttp://en.wikipedia.org/wiki/Earthhttp://en.wikipedia.org/wiki/Gravitational_constant


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FREE FALL

Free fall is any motion of a body where gravity is the only forceacting upon it, at least initially. These conditions producean inertial trajectory so long as gravity remains the only force.Since this definition does not specify velocity, it also applies to

objects initially moving upward. Since free fall in the absence offorces other than gravity producesweightlessness or "zero-g,"sometimes any condition of weightlessness due to inertial motion isreferred to as free-fall. This may also apply to weightlessnessproduced because the body is far from a gravitating body.

Near the surface of the Earth, an object in free fall in a vacuumwill accelerate at approximately 9.8 m/s2, independent of its mass.With air resistance acting upon an object that has been dropped, theobject will eventually reach a terminal velocity, around 56 m/s(200 km/h or 120 mph) for a human body. Terminal velocity depends onmany factors including mass, drag coefficient, and relative surfacearea and will only be achieved if the fall is from sufficient

altitude..

Free fall was demonstrated on the moon by astronaut DavidScott on August 2, 1971. He simultaneously released a hammer and afeather from the same height above the moon's surface. The hammer andthe feather both fell at the same rate and hit the ground at the sametime. This demonstrated Galileo's discovery that in the absence of airresistance, all objects experience the same acceleration due togravity. (On the Moon, the gravitational acceleration is much lessthan on Earth, approximately 1.6 m/s2).

Uniform gravitational field without air resistance

whereis the initial velocity (m/s).

is the vertical velocity with respect to time (m/s).

is the initial altitude (m).

is the altitude with respect to time (m).

is time elapsed (s).

is the acceleration due to gravity (9.81 m/s2

near the surfaceof the earth).

Uniform gravitational field with air resistance
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Assuming an object falling from rest and no change in air densitywith altitude, the solution is:

where the terminal speedis given by

The object's speed versus time can be integrated over time tofind the vertical position as a function of time:

Inverse-square law gravitational field

It can be said that two objects in space orbiting each other inthe absence of other forces are in free fall around each other,e.g.that the Moon or an artificial satellite "falls around" the Earth,or a planet "falls around" the Sun. Assuming spherical objects meansthat the equation of motion is governed by Newton's Law of UniversalGravitation, with solutions to the gravitational two-bodyproblembeingelliptic orbits obeying Kepler's laws of planetarymotion. This connection between falling objects close to the Earth andorbiting objects is best illustrated by the thoughtexperimentNewton's cannonball.

The motion of two objects moving radially towards each other withno angular momentum can be considered a special case of an elliptical

orbit of eccentricity e = 1 (radial elliptic trajectory). This allowsone to compute the free-fall time for two point objects on a radialpath. The solution of this equation of motion yields time as afunction of separation:

wheret is the time after the start of the fally is the distance between the centers of the bodiesy0 is the initial value of y

= G(m1 + m2) is the standard gravitational parameter.Substituting y=0 we get the free-fall time.

The separation as a function of time is given by the inverse ofthe equation. The inverse is represented exactly by the analytic powerseries:
http://en.wikipedia.org/wiki/Terminal_speedhttp://en.wikipedia.org/wiki/Gravitational_two-body_problemhttp://en.wikipedia.org/wiki/Gravitational_two-body_problemhttp://en.wikipedia.org/wiki/Elliptic_orbitshttp://en.wikipedia.org/wiki/Kepler%27s_laws_of_planetary_motionhttp://en.wikipedia.org/wiki/Kepler%27s_laws_of_planetary_motionhttp://en.wikipedia.org/wiki/Newton%27s_cannonballhttp://en.wikipedia.org/wiki/Orbital_eccentricityhttp://en.wikipedia.org/wiki/Radial_elliptic_trajectoryhttp://en.wikipedia.org/wiki/Free-fall_timehttp://en.wikipedia.org/wiki/Standard_gravitational_parameterhttp://en.wikipedia.org/wiki/Free-fall_timehttp://en.wikipedia.org/wiki/Free-fall_timehttp://en.wikipedia.org/wiki/Standard_gravitational_parameterhttp://en.wikipedia.org/wiki/Free-fall_timehttp://en.wikipedia.org/wiki/Radial_elliptic_trajectoryhttp://en.wikipedia.org/wiki/Orbital_eccentricityhttp://en.wikipedia.org/wiki/Newton%27s_cannonballhttp://en.wikipedia.org/wiki/Kepler%27s_laws_of_planetary_motionhttp://en.wikipedia.org/wiki/Kepler%27s_laws_of_planetary_motionhttp://en.wikipedia.org/wiki/Elliptic_orbitshttp://en.wikipedia.org/wiki/Gravitational_two-body_problemhttp://en.wikipedia.org/wiki/Gravitational_two-body_problemhttp://en.wikipedia.org/wiki/Terminal_speed


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Evaluating this yields:

REFERENCES

www.ngs.noaa.gov/PUBS_LIB/Geodesy4Layman/TR80003D.HTM Geodesy for the Layman. Defense Mapping Agency http://www.ipgp.fr/~tarantola/Files/Professional/Teaching/Seminar

/Texts/Barlier-Lefebvre.pdf

www.ulb.tu-darmstadt.de/tocs/112785794.pdf
http://www.ngs.noaa.gov/PUBS_LIB/Geodesy4Layman/TR80003D.HTMhttp://www.ngs.noaa.gov/PUBS_LIB/Geodesy4Layman/TR80003D.HTMhttp://www.ngs.noaa.gov/PUBS_LIB/Geodesy4Layman/TR80003D.HTMhttp://www.ngs.noaa.gov/PUBS_LIB/GeoLay.pdfhttp://www.ipgp.fr/~tarantola/Files/Professional/Teaching/Seminar/Texts/Barlier-Lefebvre.pdfhttp://www.ipgp.fr/~tarantola/Files/Professional/Teaching/Seminar/Texts/Barlier-Lefebvre.pdfhttp://www.ipgp.fr/~tarantola/Files/Professional/Teaching/Seminar/Texts/Barlier-Lefebvre.pdfhttp://www.ipgp.fr/~tarantola/Files/Professional/Teaching/Seminar/Texts/Barlier-Lefebvre.pdfhttp://www.ipgp.fr/~tarantola/Files/Professional/Teaching/Seminar/Texts/Barlier-Lefebvre.pdfhttp://www.ipgp.fr/~tarantola/Files/Professional/Teaching/Seminar/Texts/Barlier-Lefebvre.pdfhttp://www.ngs.noaa.gov/PUBS_LIB/GeoLay.pdfhttp://www.ngs.noaa.gov/PUBS_LIB/Geodesy4Layman/TR80003D.HTM

The First Satellites

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