satellite motion notes

8/10/2019 Satellite Motion Notes

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THE BIG

IDEA ........

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SATELLITE MOTION

I

f you drop a stone, it will

fall in a straight-line path to

the ground below. If you moveyour hand horizontally as you drop

the stone, it will follow a curved path

to the ground. If you move your hand

faster, the stone will land farther away and

the curvature of the path will be less pro-

nounced. What would happen if the curvature

of the path matched the curvature of Earth?

The answer is simple enough: Without air

resistance, youd have an Earth satellite!

The path of an Earth satellitefollows the curvature of the Earth.

41

What Happens When You Disturb thePath of a Pendulum?

1. Make a pendulum from a mass and a 1-m

long string. Tie the free end of the string to asupport.

2. Set the pendulum swinging. It should moveback and forth only and not side-to-side.

3. While the pendulum is swinging back andforth, tap the mass sideways.

4. Repeat Step 3 several times, each time tap-ping the mass with a different force.

Analyze and Conclude

1. Observing Describe the original shape of thepath of the mass.

2. Drawing Conclusions What effect does tap-ping the mass with different forces have onthe shape of the path?

3. Predicting How might changing the amountof mass affect the nature of the path?

discover!


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CHAPTER 14 SATELLITE MOTION 26

14.1 Earth SatellitesSimply put, an Earth satellite is a projectile moving fast enough tofall continually aroundEarth rather than intoit. Imagine yourself ona planet that is smaller than Earth as shown in Figure 14.2. Becauseof the planets small size and low mass, you would not have to throwthe stone very fast to make its curved path match the surface curva-ture of the planet. If you threw the stone just right, it would follow acircular orbit.

How fast would the stone have to be thrown horizontally for it to

orbit Earth? The answer depends on the rate at which the stone fallsand the rate at which Earth curves. Recall from Chapter 4 that a stonedropped from rest accelerates downward (or toward the center ofEarth) at 10 m/s2and falls a vertical distance of 5 meters during thefirst second. Also recall from Chapter 5 that the same is true of anyprojectile as it starts to fall. Recall that in the first second a projectilewill fall a vertical distance of 5 meters below the straight-line path itwould have taken without gravity as shown in Figure 14.3. (It maybe helpful to refresh your memory and review Figure 5.11.)

5 m 5 m5 m

FIGURE 14.1

The greater the stoneshorizontal motion whenreleased, the wider thearc of its curved path.

FIGURE 14.2

If you toss the stone horizon-tally with the proper speed,its path will match the sur-face curvature of the smallplanet.

FIGURE 14.3

Throw a stone at any speedand one second later it wilhave fallen 5 m below wheit would have been withougravity.


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A geometric fact about the curvature of our Earth is that itssurface drops a vertical distance of nearly 5 meters for every8000 meters tangent to its surface as shown in Figure 14.4.

A stone thrown fast enough to go a horizontal distance of8000 meters during the time (1 second) it takes to fall 5 meters,will orbit Earth.Isnt this speed simply 8000 meters per second?So we see that the orbital speed for close orbit about Earth is8000 m/s (or 8 km/s). If this doesnt seem to be very fast, convertit to kilometers per hour; youll see it is an impressive 29,000 km/h(or 18,000 mi/h). At that speed, atmospheric friction would burn an

object to a crisp. Thats why a satellite must stay about 150 kilometersor more above Earths surfaceto keep from burning due to the fric-tion of the atmosphere.

CONCEPT

CHECK ......Near the surface of Earth, how fast does a stone

have to be thrown to orbit Earth?

14.2 Circular OrbitsInterestingly, in circular orbit the speed of a circling satellite is notchanged by gravity. We can understand this by comparing a satel-lite in circular orbit to a bowling ball rolling along a bowling alley asshown in Figure 14.5. Why doesnt the gravity that acts on the bowl-ing ball change its speed? The answer is that gravity is pulling neitherforward nor backwardit pulls straight downward, perpendicularto the balls motion. The bowling ball has no component of gravita-tional force along the direction of the alley.

5 m

8000 m FIGURE 14.4

In the curvature of Earth,the surface drops a verticadistance of nearly 5 meterfor every 8000 meters tangent to its surface.

The 5-meter drop foreach 8000-meter tan-gent means that if youwere floating in a calmocean youd be able tosee only the top of a5-meter mast on a boat8000 meters away.

FIGURE 14.5

The speeds of the bowling

ball and the satellite arenot affected by the force ogravity because there is nohorizontal component ofthe gravitational force.


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266

The same is true for a satellite in circular orbit. Here a satellite isalways moving at a right angle (perpendicular) to the force of grav-ity. It doesnt move in the direction of gravity, which would increaseits speed, nor does it move in a direction against gravity, whichwould decrease its speed. Instead, the satellite exactly criss-crossesgravity, so that no change in speed occursonly a change in direc-

tion. A satellite in circular orbit around Earth is always movingperpendicular to gravity and parallel to Earths surface at constantspeed.

For a satellite close to Earth, the time for a complete orbitaround Earth, its period, is about 90 minutes. For higher altitudes,the orbital speed is less and the period is longer. Communicationssatellites are located in orbit 6.5 Earth radii from Earths center, sothat their period is 24 hours. This period matches Earths daily rota-tion. They are launched to orbit in the plane of Earths equator, sothey are always above the same place on the equator. The moon is

farther away, and has a 27.3-day period. The higher the orbit of asatellite, the slower its speed and the longer its period.14.2

The international space station (ISS), shown in Figure 14.6,orbits at 360 kilometers above Earths surface. Like all satellites, tan-gential velocity assures that it falls around Earth rather than intoit. Acceleration toward Earth is somewhat less than 1gbecause ofaltitude. This acceleration, however, is not sensed by the astronauts;

relative to the station, they experience zerog. Over extended periodsof time this causes loss of muscle strength and other detrimentalchanges in the body. Future space travelers, however, can avoid thiswhen space stations rotate (recall our discussion of rotating spacehabitats in Chapter 12). Rotation effectively supplies a support forceand can nicely provide Earth-normal weight.

think!

FIGURE 14.6

The ISS and its inhabitantscircle 360 km above theEarth, well above its atmo-sphere, in a state of con-tinual free fall.

Satellites in close circularorbit fall about 5 m dur-ing each second of orbit.How can this be if the sat-ellite does not get closer

to Earth?Answer: 14.2

Think of the InternationalSpace Station as Earthslifeboat.


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Recall from Chapter 13 that Isaac Newton understood satellitemotion from his investigation of the moons motion. He thoughtabout the launching of artificial satellites, for he reasoned that with-out air resistance, a cannonball could circle Earth and coast indefi-nitely if it had sufficient speed. As Figure 14.7 demonstrates, hecalculated this speed to be the same as 8 km/s. Since such speed wasimpossible then, he was not optimistic about people launching satel-lites. What Newton did not consider was multistage rocketsthe ideaof rockets carried piggyback style on other rockets to reach orbitalspeed by a succession of rocket firings.

CONCEPT

CHECK ......Describe the motion of a satellite in relation to

Earths surface and gravity.

14.3 Elliptical OrbitsA satellite in orbit around Earth traces an oval-shaped path

called an ellipse.An ellipse is the closed path taken by a point thatmoves in such a way that the sum of its distances from two fixed

points is constant. The two fixed points in an ellipse are called foci.For a satellite orbiting a planet, the center of the planet is at onefocus and the other focus could be inside or outside the planet. Anellipse can be easily constructed by using a pair of tacks, one at eachfocus, a loop of string, and a pencil, as shown in Figure 14.8. Thecloser the tacks, the closer the ellipse is to a circle. When thefoci are together, the ellipse isa circle. A circle is a special caseof an ellipse.

How Do You Draw Different Ellipses?

1. Draw an ellipse with a loop of string, two tacks, and a pen orpencil, as shown in Figure 14.8.

2. Try different tack spacings for a variety of ellipses. Or draw anellipse by tracing the edge of the shadow cast by a circular disk ona flat surface.

3. Think How can you move the disk to get different ellipses?

discover!

FIGURE 14.7

A satellite in circular orbit close toEarth moves tangentially at 8 km/s.Each second, it falls 5 m beneatheach successive 8-km tangent.

When a spacecraftenters the atmosphereat too steep an angle,more than about6 degrees, it can burnup. If it comes in tooshallow it could bounceback into space like apebble skipped acrosswater.

FIGURE 14.8

A simple method ofconstructing an ellipseis shown here.


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The orbit of a satellite isshown in the sketch. Inwhich of the positions Athrough Ddoes the sat-

ellite have the greatestspeed? The least speed?Answer: 14.3

think! Satellite speed, which is constant in a circular orbit, varies in anelliptical orbit. When the initial speed is more than 8 km/s, the satelliteovershoots a circular path and moves away from Earth, against theforce of gravity. It therefore loses speed. Like a rock thrown into theair, the satellite slows to a point where it no longer recedes, and begins

falling back toward Earth. The speed lost in receding is regained as itfalls back. The satellite then rejoins its path with the same speed it hadinitially. The procedure repeats over and over, and an ellipse is tracedeach cycle as shown in Figure 14.9.

Quite interestingly, the apparently parabolic paths of projectilessuch as cannonballs are actually tiny segments of a thin ellipse thatextends within and just beyond the center of Earth. Figure 14.10shows that for speeds less than orbital speeds, the center of Earth isthe far focus of the elliptical path. In this case the near focus is closeto the launching site and varies for different speeds. When the projec-

tile traces a circular orbit, both foci are together at Earths center. Forelliptical orbits, the near focus is Earths center and the location ofthe far focus varies for different speeds.

CONCEPT

CHECK ......What is the shape of the path of a satellite in an

orbit around Earth?

ba

a b c

FIGURE 14.9

A satellite moves in an ellipti-cal orbit. a.When the satelliteexceeds 8 km/s, it overshootsa circle. b.At its maximumseparation, it starts to comeback toward Earth. c.The

cycle repeats itself.

FIGURE 14.10

The parabolic paths of projectilesare actually segments of ellipses.

a.For relatively low speeds, thecenter of Earth is the far focus.b.For greater speeds, the nearfocus is Earths center.


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14.4 Energy Conservation andSatellite Motion

Recall from Chapter 9 that moving objects have kinetic energy (KE).

An object above Earths surface has potential energy (PE) due to itsposition. Everywhere in its orbit, a satellite has both KE and PE.

The sum of the KE and PE of a satellite is constant at all pointsalong an orbit.

In a circular orbit, the distance between a planets center and thesatellites center is constant, as shown in Figure 14.11. This means thatthe PE of the satellite is the same everywhere in orbit. So, by the law ofconservation of energy, the KE is also constant. Thus, the speed is con-stant in any circular orbit.

In an elliptical orbit the situation is different. Both speed and

distance vary. The apogee is the point in a satellites orbit farthestfrom the center of Earth. The perigee is the point in a satellitesorbit closest to the center of Earth. The PE is greatest when the sat-ellite is at the apogee and least when the satellite is at the perigee.Correspondingly, the KE will be least when the PE is most; and theKE will be most when the PE is least, as Figure 14.12 shows. At everypoint in the orbit, the sum of the KE and PE is constant.

Satellite Design Engineer Satellites play an important role inconducting scientific research, obtaining environmental data, andproviding communications services. Satellite design engineers areemployed by the United States government through NASA and bycommercial communications companies. The goal of a satellite designengineer is to design satellites that will orbit at specific distances fromEarth, carry the necessary equipment, and withstand the conditions towhich they will be exposedall of this within a controlledmonetary budget.

Physics on the Job

FIGURE 14.11

For a satellite in circular orbitno component of force actsalong the direction of motionThe speed, and thus the KE,cannot change.

FIGURE 14.12

The sum of KE and PEfor a satellite is a con-stant at all points alongan elliptical orbit.


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270

The orbital path of asatellite is shown below.In which of the positionsAthrough Ddoes thesatellite have the mostKE? Most PE? Most totalenergy? Answer: 14.4

At all points on the orbitexcept at the apogee and perigeethere is a component of gravitational force parallel to the directionof satellite motion, as Figure 14.13 shows. This component changes

the speed of the satellite. Or we can say: (this component of force)

(distance moved) = change in KE. Either way we look at it, when thesatellite gains altitude and moves against this component, its speedand KE decrease. The decrease continues to the apogee. Once pastthe apogee, the satellite moves in the same direction as the compo-nent, and the speed and KE increase. The increase continues untilthe satellite whips past the perigee and repeats the cycle.

CONCEPT

CHECK ......What is the relationship between the KE and PE of asatellite in motion?

14.5 Keplers Laws Of PlanetaryMotion

Newtons law of gravitation was preceded by Keplers laws of plan-etary motion. Keplers laws of planetary motion are three impor-tant discoveries about planetary motion that were made by theGerman astronomer Johannes Kepler in the beginning of the 1600s.Keplers career as an astronomer began with a junior assistantshipwith the famed Danish astronomer Tycho Brahe. Brahe headed theworlds first great observatory in Denmark, just prior to the adventof the telescope. Using huge brass protractor-like instruments calledquadrants, Brahe measured the positions of planets over twentyyears so accurately that his measurements are still valid today. Braheentrusted his data to Kepler. After Brahes death, Kepler devotedmany years of his life to the analysis of Brahes measurements.

think!

FIGURE 14.13

In an elliptical orbit, acomponent of force existsalong the direction of thesatellites motion. This

component changes thespeed and, thus, the KE.


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Keplers First Law Keplers expectation that the planets wouldmove in perfect circles around the sun was shattered after years ofeffort. He found the paths to be ellipses. Keplers first law statesthat the path of each planet around the sun is an ellipse with thesun at one focus.

Keplers Second Law Kepler also found that the planets do notgo around the sun at a uniform speed but move faster when they arenearer the sun and more slowly when they are farther from the sun.They accomplish this in such a way that an imaginary line or spokejoining the sun and the planet sweeps out equal areas of space inequal intervals of time. The triangular-shaped areas swept out dur-ing a month when a planet is orbiting far from the sun and when aplanet is orbiting closer to the sun are shown in Figure 14.15. These

two areas are equal. Keplers second law states that each planetmoves so that an imaginary line drawn from the sun to any planetsweeps out equal areas of space in equal intervals of time.

Kepler was the first to coin the word satellite. He had no clearidea whythe planets moved as he discovered. He lacked a concep-tual model. Kepler didnt see that a satellite is simply a projectileunder the influence of a gravitational force directed toward the bodyaround which the satellite orbits. You know that if you toss a rockupward, it goes slower the higher it rises because its going againstEarth gravity. And you know that when it returns its going with

gravity and its speed increases. Kepler never realized that a satellitebehaves in the same way. Going away from the sun, it slows down.Returning toward the sun, it speeds up. A satellite, whether a planetorbiting the sun, or one of todays satellites orbiting Earth, is slowedgoing against the gravitational field and sped up going with the field.Kepler wasnt aware of this simplicity, and instead fabricated complexsystems of geometrical figures to find sense in his discoveries. Theseproved futile.

a.Tycho Brahe b. Johannes Kepler

FIGURE 14.14

a.Tycho Brahe (15461601)measured the positions ofplanets over 20 years soaccurately that his measure-ments are still valid today.

b.Johannes Kepler(15711630) devoted manyyears of his life to the analysisof Brahes measurements.

FIGURE 14.15

Equal areas of space areswept out in equal inter-vals of time.

Keplers second law isa consequence of theconservation of angularmomentum. And his

third law is the result ofequating Newtons lawof gravitation to centripetal force. The connec-tions of conceptsyum


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Keplers Third Law After ten years of searching by trial anderror for a connection between the time it takes a planet to orbitthe sun and its distance from the sun, Kepler discovered a thirdlaw. From Brahes data, Kepler found that the square of any planetsperiod (T) is directly proportional to the cube of its average orbitalradius (r). Keplers third law states that the square of the orbital

period of a planet is directly proportional to the cube of the aver-age distance of the planet from the sun.This means that the ratioT2/r3 is the same for all planets. So if a planets period is known, itsaverage orbital radial distance is easily calculated (or vice versa).Keplers laws apply not only to planets but also to moons or anysatellite in orbit around any body. The elliptical orbits of the planetsare very nearly circular. Only the precise measurements of Braheshowed the slight differences.

It is interesting to note that Kepler was familiar with Galileosconcepts of inertia and accelerated motion, but he failed to apply

them to his own work. Like Aristotle, he thought that the force on amoving body would be in the same direction as the bodys motion.Kepler never appreciated the concept of inertia. Galileo, on the otherhand, never appreciated Keplers work and held to his convictionthat the planets move in circles.14.6Further understanding of plan-etary motion required someone who could integrate the findings ofthese two great scientists. The rest is history, for as we have seen, thistask was later taken up by Isaac Newton.

CONCEPT

CHECK ......

What are Keplers three laws of planetary motion?

14.6 Escape SpeedWhen a payload is put into Earth-orbit by a rocket, the speed anddirection of the rocket are very important. For example, what wouldhappen if the rocket were launched vertically and quickly achieveda speed of 8 km/s? Everyone had better get out of the way, because itwould soon come crashing back at 8 km/s. As Figure 14.16 shows, toachieve orbit, the payload must be launched horizontallyat 8 km/sonce above air resistance. Launched vertically, the old saying Whatgoes up must come down becomes a sad fact of life.

Earth But isnt there some vertical speed that is sufficient to ensurethat what goes up will escape and not come down? The answer is yes.Neglecting air resistance, fire anything at any speed greater than11.2 km/s, and it will leave Earth, going more and more slowly, butnever stopping.14.6.1Lets look at this from an energy point of view.

The mass of any celes-tial body can be foundif it has one or moresatellites, for the bodysmass is directly propor-tional to r3/T2.

FIGURE 14.16

The initial thrust of therocket lifts it vertically.Another thrust tips it fromits vertical course. Whenit is moving horizontally, itis boosted to the requiredspeed for orbit.


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How much work is required to move a payload against the forceof Earths gravity to a distance very, very far (infinitely far) away?The PE is not infinite because the distance is infinite. But gravitydiminishes rapidly with distance via the inverse-square law. Most ofthe work done in launching a rocket, for example, occurs near Earth.It turns out that the value of PE for a 1-kilogram mass infinitely far

away is 62 million joules (MJ). So to put a payload infinitely far fromEarths surface requires at least 62 MJ of energy per kilogram of load.A KE per unit mass of 62 MJ/kg corresponds to a speed of 11.2 km/s.This is the value of the escape speed from the surface of Earth.14.6.2

The escape speed is the minimum speed necessary for an object toescape permanently from a gravitational field that holds it.

If we give a payload any more energy than 62 MJ/kg at thesurface of Earth or, equivalently, any greater speed than 11.2 km/s,then, neglecting air resistance, the payload will escape from Earthnever to return. As it continues outward, its PE increases and its KE

decreases. Its speed becomes less and less, though it is never reducedto zero. The payload outruns the gravity of Earth. It escapes.

The Solar System The escape speeds of various bodies in thesolar system are shown in Table 14.1. Note that the escape speed fromthe sun is 620 km/s at the surface of the sun. Even at a distance equal-ing that of Earths orbit, the escape speed from the sun is 42.2 km/s.The escape speed values in the table ignore the forces exerted by otherbodies. A projectile fired from Earth at 11.2 km/s, for example, escapesEarth but not necessarily the moon, and certainly not the sun.

Escape Speeds at the Surface of Bodies in the Solar SystemTable 14.1

Sun

Jupiter

Saturn

Neptune

Uranus

Earth

Venus

Mars

Mercury

Moon

AstronomicalBody

Mass(Earth masses)

Radius(Earth radii)

Escape Speed(km/s)

Sun (at a distanceof Earths orbit)

620

42.2

60.2

36.0

24.9

22.3

11.2

10.4

5.0

4.3

2.4

109

23,500

11

9.2

3.47

3.7

1.00

0.95

0.53

0.38

0.28

333,000

333,000

318

95.2

17.3

14.5

1.00

0.82

0.11

0.055

0.0123


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The first probe to escape the solar system, Pioneer 10,shown inFigure 14.17, was launched from Earth in 1972 with a speed of only15 km/s. The escape was accomplished by directing the probe into thepath of oncoming Jupiter. It was whipped about by Jupiters great

gravitational field, picking up speed in the processjust as the speedof a ball encountering an oncoming bat is increased when it departsfrom the bat. Its speed of departure from Jupiter was increasedenough to exceed the suns escape speed at the distance of Jupiter.Pioneer 10passed the orbit of Pluto in 1984. Unless it collides withanother body, it will continue indefinitely through interstellar space.Like a note in a bottle cast into the sea, Pioneer 10 contains informa-tion about Earth that might be of interest to extraterrestrials, in hopesthat it will one day wash up and be found on some distant seashore.

It is important to point out that the escape speeds for different

bodies refer to the initial speed given by a brief thrust, after whichthere is no force to assist motion. But we could escape Earth atany sustainedspeed greater than zero, given enough time. Supposea rocket is going to a destination such as the moon. If the rocketengines burn out while still close to Earth, the rocket will need a min-imum speed of 11.2 km/s. But if the rocket engines can be sustainedfor long periods of time, the rocket could go to the moon withoutever attaining 11.2 km/s.

It is interesting to note that the accuracy with which an unpi-loted rocket reaches its destination is accomplished not by staying

on a preplanned path, or by getting back on that path if it straysoff course. No attempt is made to return the rocket to its plannedpath. Instead, by communication with the control center, the rocketin effect asks, Where am I now, and where do I want to go? Whatis the best way to get there from here, given my present situation?With the aid of high-speed computers, the answers to these ques-tions are used to find a new path. Corrective thrusters put the rocketon this new path. This process is repeated continuously along theway until the rocket reaches its destination.

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FIGURE 14.17

Pioneer 10, launched fromEarth in 1972, escaped fromthe solar system in 1984 and iswandering in interstellar space.

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Is there a lesson to be learned here? Suppose you find in yourpersonal life that you are off course. You may, like the rocket, findit better to take a newer course that leads to your goal as best plottedfrom your present position and circumstances, rather than try to getback on the course you plotted from a previous position and in, per-haps, different circumstances. So many ideas in physics, it seems, have

a moral.

CONCEPT

CHECK ......What condition is necessary for a payload to escape

Earths gravity?

Communications Satellites The electromagneticsignals that are broadcast into space to carrytelevision programs or telephone conversationstravel in straight lines. In times past these straight-line (often called line-of-sight) communicationsrequired tall receiving antenna towers and signal-boosting relay stations on high buildings ormountains. Today many television and telephonesignals bounce to usfrom satellites. Thesecommunications satellitesare in equatorial orbitswith 24-hour periods.Because they revolveonce each time Earthrotates once, they appearstationary when welook up at them. Thesesatellites are said to be ingeosynchronous orbits.

Dish-shaped antennas almost anywhere on Earth areon a line of sight from one or more communicationssatellites. Because communications satellites are inequatorial orbit, dish antennas on the equator maytilt east or west, but they dont tilt north or south.An equatorial dish right under a communicationssatellite is looking straight up. If it held water, itwould resemble a birdbath filled to the brim.

Antennas north of the equator must be tilted tothe south (and perhaps east or west, too). Thosesouth of the equator must be tilted to the north,and likely east or west as well. Unless you live onthe equator, all of the antennas that you see looklike partly emptied bowls. In Antarctica or near theNorth Pole, a dish antenna is tipped so far overthat it could hold no water at all.

The fact thatgeosynchronous satellitesremain in one place

overhead means it ispossible for them todrop vertical cables toEarths surface wherethey could be attached.Cables composed of verystrong and lightweightcarbon-based materials arecurrently being researched.

Rather than being rocketed to the satellite,supplies could be lifted in elevator fashion. Watchfor such space elevators in the future!

Critical Thinking Explain why it is not possible fora single communications satellite to serve all partsof Earth.

Science, Technology, and Society

The mind that encom-passes the universe isas marvelous as the uni-verse that encompassesthe mind.


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Concept Summary

A stone thrown fast enough to go a hori-zontal distance of 8000 meters during thetime (1 second) it takes to fall 5 meterswill orbit Earth.

A satellite in circular orbit around Earthis always moving perpendicular to gravityand parallel to Earths surface at constant

speed.

A satellite in orbit around Earth tracesand oval-shaped path called an ellipse.

The sum of the KE and PE of a satellite isconstant at all points along an orbit.

Keplers first law states that the path ofeach planet around the sun is an ellipsewith the sun at one focus.

Keplers second law states that eachplanet moves so that an imaginary linedrawn from the sun to any planet sweepsout equal areas of space in equal intervalsof time.

Keplers third law states that the square ofthe orbital period of a planet is directlyproportional to the cube of the averagedistance of the planet from the sun.(T2~ r3for all planets)

If we give a payload any more energy

than 62 MJ/kg at the surface of Earth or,equivalently, any greater speed than11.2 km/s, then, neglecting air resistance,the payload will escape from Earth neverto return.

14.2 In each second, the satellite falls about

5 m below the straight-line tangent it

would have taken if there were no gravity.

Earths surface curves 5 m below an 8-km

straight-line tangent. Since the satellite

moves at 8 km/s, it falls at the same rate

Earth curves.

14.3 The satellite has its greatest speed as it

whips aroundA.It has its least speed at C.

Beyond C,it gains speed as it falls back to

Ato repeat its cycle.

14.4 The KE is maximum atA; the PE is maxi-

mum at C; the total energy is the same

anywhere in the orbit.

think! Answers

satellite(p. 263)

period(p. 266)

ellipse(p. 267)

focus (pl.foci)(p. 267)

apogee(p. 269)

perigee(p. 269)Keplers laws of planetary motion(p. 270)

escape speed(p. 273)

Key Terms


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CHAPTER 14 SATELLITE MOTION 27CHAPTER 14 SATELLITE MOTION 27

41 ASSESSCheck Concepts

Section 14.1

1. If we drop a ball from rest, how far willit fall vertically in the first second? If we in-stead move our hand horizontally and dropit (throw it), how far will it fall vertically inthe first second?

2. What do the distances 8000 m and5 m have to do with a line tangent to

Earths surface?

Section 14.2

3. How does the direction of motion of asatellite in circular orbit compare with thecurve of Earths surface?

4. Why doesnt gravitational force changethe speed of a satellite in circular orbit?

5. Does the period of a satellite in a circu-lar orbit increase or decrease as its distancefrom Earth increases?

Section 14.3

6. Describe an ellipse.

Section 14.4

7. Why does gravitational force change thespeed of a satellite in elliptical orbit?

8. a. Where in an elliptical orbit is the speed oa satellite maximum?

b. Where is it minimum?

9. The sum of PE and KE for a satellite ina circular orbit is constant. Is this sum alsoconstant for a satellite in an elliptical orbit?

10. Why does the force of gravity do nowork on a satellite in circular orbit, but doe

do work on a satellite in an elliptical orbit?

Section 14.5

11.What scientist gathered accurate data onplanetary paths around the sun? What sci-entist discovered that the paths are ellipses?What scientist explained the ellipses?

12.When is the speed of a satellite greatest,when closer to Earth or farther from Earth?

13.What is the mathematical relationship be-tween how long it takes a planet to orbit thsun and its distance from the sun?


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csa 1400ASSESSASSESS (continued)

278

41Section 14.6

14. a. What is the minimum speed forcircling Earth in close orbit?

b. What is the maximum speed in an orbitthat comes close to Earth at one point?

c. What happens above the maximumspeed?

15. Neglecting air resistance, what will hap-

pen to a projectile that is fired vertically at8 km/s? At 12 km/s?

16. a. How fast would a particle have to beejected from the sun to leave the solarsystem?

b. What speed would be needed if an ejectedparticle started at a distance from the sunequal to Earths distance from the sun?

17. What is the escape speed on the moon?

18. Although the escape speed from the sur-face of Earth is 11.2 km/s, couldnt a rocketwith enough fuel escape at any speed? Whyor why not?

19. How was Pioneer 10able to escape thesolar system with an initial speed less thanescape speed?

Think and Rank

Rank each of the following sets of scenarios inorder of the quantity or property involved. Listthem from left to right. If scenarios have equalrankings, then separate them with an equal sign.(e.g., A = B)

20.The dashed lines show three circular orbitsabout Earth.

Rank the following quantities for theseorbits from greatest to least.

a. orbital speed

b. time for orbiting Earth

21.Four satellites in circular orbit about Earthhave the following characteristics:

(A)m 4000 kg; height 300 km(B) m 5000 kg; height 350 km(C)m 4000 kg; height 400 km(D)m 5000 kg; height 500 km

a. Rank the satellites orbital speeds from

greatest to least. b. Rank the satellites times for orbitingEarth from greatest to least.

c. Does mass affect your answers to parts (a)and (b)?


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41 ASSESS


22.The positions of a satellite in elliptical orbitare indicated.

Rank these quantities from greatest to least.a. gravitational force

b. speed c. momentum d. KE e. PE f. total energy (KE PE)

g. acceleration

23.Kepler tells us that a planet sweeps out equalareas in equal intervals of time. Four suchequal-area triangles are shown.

Rank these quantities from greatest to least.a. average speed during the time interval

b. acceleration during the time interval

Think and Explain

24. A satellite can orbit at 5 km above themoon, but not at 5 km above Earth. Why?

25. Does the speed of a satellite aroundEarth depend on its mass? Its distance fromEarth? The mass of Earth?

26. If a cannonball is fired from a tallmountain, gravity changes its speed allalong its trajectory. But if it is fired fastenough to go into circular orbit, gravitydoes not change its speed at all. Why?

27.Does gravity do any network on a satellite

in an elliptical orbit during one full orbit?Explain your answer.

28.A geosynchronous Earth satellite can remaialmost directly overhead in Singapore, butnot San Francisco, Chicago, or New YorkCity. Why?

29. If you stopped an Earth satellite dead inits tracks, it would simply crash into Earth.Why, then, dont the communications satel

lites that hover motionless above the samespot on Earth crash into Earth?


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csa 1400ASSESSASSESS (continued)

280

4136. Why does most of the work done in

launching a rocket take place when therocket is still close to Earths surface?

37. If Pluto were somehow stopped shortin its orbit, it would fall into the sun ratherthan around it. About how fast would it bemoving when it hit the sun?

38. If an astronaut in an orbiting spaceshuttle wished to drop something to Earth,

how could this be accomplished?39. If Earth somehow acquired more mass,

with no change in its radius, would escapespeed be less than, equal to, or more than11.2 km/s? Why?

Think and Solve

40. Calculate the speed in m/s at whichEarth revolves around the sun. Note: The

orbit is nearly circular.

41. A spaceship in circular orbit about themoon is 2.0 106m from its center.a. Show that the period of the spaceship is

2.2 h.b. Show that the speed of the spaceship rela-

tive to the moon is about 5800 km/h.

30.In an accidental explosion, a satellite breaksin half while in circular orbit about Earth.One half is brought momentarily to rest.What is the fate of the half brought to rest?What is the speed of the other half? (Hint:Think momentum conservation.)

31. Would you expect the speed of a satellitein close circular orbit about the moon to be

less than, equal to, or greater than 8 km/s?Why?

32. Why do you suppose that sites closeto the equator are preferred for launchingsatellites? (Hint:Look at the spinning Earthfrom above either pole and compare it to aspinning turntable.)

33. Why do you suppose that a space shuttleis sent into orbit by firing it in an easterly

direction (the direction in which Earthspins)?

34.Consider two planets: Mercury, close to thesun, and Uranus, far from the sun. Whichof these planets has a period shorter thanEarths period around the sun? Which has aperiod longer than Earths?

35. What is the maximum possible speed ofimpact upon Earths surface for a faraway

object initially at rest that falls to Earth dueonly to Earths gravity?


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41 ASSESS


More Problem-Solving PracticeAppendix F

42. Calculate the speed in m/s at which themoon revolves around Earth. Note: Theorbit is nearly circular.

43.At a particular point, a satellite in anelliptical orbit has a gravitationalpotential energy of 5000 MJ with respectto Earths surface and a kinetic energy of4500 MJ. At another point in its orbit, the

satellites potential energy is 6000 MJ. Whatis its kinetic energy at that point?

44. An orbiting satellite of mass mis pulledtoward Earth by a force ma. Equate matothe force in Newtons equation for universalgravitation and show that the satellites

acceleration is a .GMd2

45.The force of gravity between Earth and an

Earth satellite is given by F GmMd2

., where

mis the mass of the satellite,Mis the massof Earth, and dis the distance between

the satellite and the center of Earth. If the

satellite follows a circular orbit, the force

keeping it in orbit must be the centripetal

force, given by F .mv

r Equate the two

expressions for force to show that the speed

is v .GM

d46.Use the result of Question 45 (now with the

sun instead of Earth as the center of force)to calculate the speed in m/s at which Earthrevolves about the sun. Assume Earths orbitis nearly circular.

47.In 1610, Galileo discovered four moons ofJupiter. (Today we know that there are morthan 60!) Io, the innermost of the moonsobserved by Galileo, is 4.2 108m fromJupiters center and has a period of1.5 105s. Calculate Jupiters mass.

48.A planet in a circular orbit takes a time Ttorbit its sun at a radial distance r. In terms

of rand T, how fast is the planet moving inits orbit?

49.The speed of a satellite in a circular orbit is

given by the equation v ,GM

r where Gthe gravitational constant,Mis the mass oEarth, and ris the radial distancebetween the satellite and the center of Eart

Equate this to the other expression for

speed, v .dt Find the equation for the

time the satellite takes to completely orbitEarththe period T. Use the circumferencof the complete orbit, 2r, for the distancetraveled, and Tfor the period of rotation.

50.Use the equation T 2r3

GM to showthat the period of the space shuttle 200 kmabove Earths surface is about 90 minutes.


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