Comet Halley (Fig. P13.23) approaches the Sun towithin 0.570 AU, and its orbital period

In his 1798 experiment, Cavendish was said to have weighed the Earth. Explain this statement.

Why dont we put a geosynchronous weather satellite in orbit around the 45th parallel? Wouldnt such a satellite be more useful in the United States than one in orbit around the equator?

(a) Explain why the force exerted on a particle by a uniform sphere must be directed toward the center of the sphere. (b) Would this statement be true if the mass distribution of the sphere were not spherically symmetric? Explain.

(a) Explain why the force exerted on a particle by a uniform sphere must be directed toward the center of the sphere. (b) Would this statement be true if the mass distribution of the sphere were not spherically symmetric? Explain.

(a) At what position in its elliptical orbit is the speed of a planet a maximum? (b) At what position is the speed a minimum?

You are given the mass and radius of planet X. How would you calculate the free-fall acceleration on this planets surface?

(a) If a hole could be dug to the center of the Earth, would the force on an object of mass m still obey Equation 13.1 there? (b) What do you think the force on m would be at the center of the Earth?

Explain why it takes more fuel for a spacecraft to travel from the Earth to the Moon than for the return trip. Estimate the difference.

A satellite in low-Earth orbit is not truly traveling through a vacuum. Rather, it moves through very thin air. Does the resulting air friction cause the satellite to slow down?

Review. A student proposes to study the gravitational force by suspending two 100.0-kg spherical objects at the lower ends of cables from the ceiling of a tall cathedral and measuring the deflection of the cables from the vertical. The 45.00-m-long cables are attached to the ceiling 1.000 m apart. The first object is suspended, and its position is carefully measured. The second object is suspended, and the two objects attract each other gravitationally. By what distance has the first object moved horizontally from its initial position due to the gravitational attraction to the other object? Suggestion: Keep in mind that this distance will be very small and make appropriate approximations.

When a falling meteoroid is at a distance above the Earths surface of 3.00 times the Earths radius

The free-fall acceleration on the surface of the Moon is about one-sixth that on the surface of the Earth. The radius of the Moon is about 0.250RE (RE 5 Earths radius 5 6.37 3 106 m). Find the ratio of their average densities, rMoon/rEarth.

Review. Miranda, a satellite of Uranus, is shown in Figure P13.13a. It can be modeled as a sphere of radius 242 km and mass 6.68 3 1019 kg. (a) Find the free-fall acceleration on its surface. (b) A cliff on Miranda is 5.00 km high. It appears on the limb at the 11 oclock position in Figure P13.13a and is magnified in Figure P13.13b. If a devotee of extreme sports runs horizontally off the top of the cliff at 8.50 m/s, for what time interval is he in flight? (c) How far from the base of the vertical cliff does he strike the icy surface of Miranda? (d) What will be his vector impact velocity?

(a) Compute the vector gravitational field at a point P on the perpendicular bisector of the line joining two objects of equal mass separated by a distance 2a as shown in Figure P13.14. (b) Explain physically why the field should approach zero as r S 0. (c) Prove mathematically that the answer to part (a) behaves in this way. (d) Explain physically why the magnitude of the field should approach 2GM/r 2 as r S `. (e) Prove mathematically that the answer to part (a) behaves correctly in this limit.

Three objects of equal mass are located at three corners of a square of edge length , as shown in Figure P13.15. Find the magnitude and direction of the gravitational field at the fourth corner due to these objects.

A spacecraft in the shape of a long cylinder has a length of 100 m, and its mass with occupants is 1 000 kg. It has strayed too close to a black hole having a mass 100 times that of the Sun (Fig. P13.16). The nose of the spacecraft points toward the black hole, and the distance between the nose and the center of the black hole is 10.0 km. (a) Determine the total force on the spacecraft. (b) What is the difference in the gravitational fields acting on the occupants in the nose of the ship and on those in the rear of the ship, farthest from the black hole? (This difference in accelerations grows rapidly as the ship approaches the black hole. It puts the body of the ship under extreme tension and eventually tears it apart.)

An artificial satellite circles the Earth in a circular orbit at a location where the acceleration due to gravity is 9.00 m/s2. Determine the orbital period of the satellite.

Io, a satellite of Jupiter, has an orbital period of 1.77 days and an orbital radius of 4.22 3 105 km. From these data, determine the mass of Jupiter.

A minimum-energy transfer orbit to an outer planet consists of putting a spacecraft on an elliptical trajectory with the departure planet corresponding to the perihelion of the ellipse, or the closest point to the Sun, and the arrival planet at the aphelion, or the farthest point from the Sun. (a) Use Keplers third law to calculate how long it would take to go from Earth to Mars on such an orbit as shown in Figure P13.19. (b) Can such an orbit be undertaken at any time? Explain.

A particle of mass m moves along a straight line with constant velocity vS 0 in the x direction, a distance b from the x axis (Fig. P13.20). (a) Does the particle possess any angular momentum about the origin? (b) Explain why the amount of its angular momentum should change or should stay constant. (c) Show that Keplers second law is satisfied by showing that the two shaded triangles in the figure have the same area when tD 2 tC 5 tB 2 tA.

Plasketts binary system consists of two stars that revolve in a circular orbit about a center of mass midway between them. This statement implies that the masses of the two stars are equal (Fig. P13.21). Assume the orbital speed of each star is 0 vS 0 5 220 km/s and the orbital period of each is 14.4 days. Find the mass M of each star. (For comparison, the mass of our Sun is 1.99 3 1030 kg.)

Two planets X and Y travel counterclockwise in circular orbits about a star as shown in Figure P13.22. The radii of their orbits are in the ratio 3:1. At one moment, they are aligned as shown in Figure P13.22a, making a straight line with the star. During the next five years, the angular displacement of planet X is 90.0 as shown in Figure P13.22b. What is the angular displacement of planet Y at this moment?

Comet Halley (Fig. P13.23) approaches the Sun to within 0.570 AU, and its orbital period is 75.6 yr. (AU is the symbol for astronomical unit, where 1 AU 5 1.50 3 1011 m is the mean EarthSun distance.) How far from the Sun will Halleys comet travel before it starts its return journey?

The Explorer VIII satellite, placed into orbit November 3, 1960, to investigate the ionosphere, had the following orbit parameters: perigee, 459 km; apogee, 2 289 km (both distances above the Earths surface); period, 112.7 min. Find the ratio vp/va of the speed at perigee to that at apogee.

Use Keplers third law to determine how many days it takes a spacecraft to travel in an elliptical orbit from a point 6 670 km from the Earths center to the Moon, 385 000 km from the Earths center

Neutron stars are extremely dense objects formed from the remnants of supernova explosions. Many rotate very rapidly. Suppose the mass of a certain spherical neutron star is twice the mass of the Sun and its radius is 10.0 km. Determine the greatest possible angular speed it can have so that the matter at the surface of the star on its equator is just held in orbit by the gravitational force.

A synchronous satellite, which always remains above the same point on a planets equator, is put in orbit around Jupiter to study that planets famous red spot. Jupiter rotates once every 9.84 h. Use the data of Table 13.2 to find the altitude of the satellite above the surface of the planet.

(a) Given that the period of the Moons orbit about the Earth is 27.32 days and the nearly constant distance between the center of the Earth and the center of the Moon is 3.84 3 108 m, use Equation 13.11 to calculate the mass of the Earth. (b) Why is the value you calculate a bit too large?

Suppose the Suns gravity were switched off. The planets would leave their orbits and fly away in straight lines as described by Newtons first law. (a) Would Mercury ever be farther from the Sun than Pluto? (b) If so, find how long it would take Mercury to achieve this passage. If not, give a convincing argument that Pluto is always farther from the Sun than is Mercury.

A satellite in Earth orbit has a mass of 100 kg and is at an altitude of 2.00 3 106 m. (a) What is the potential energy of the satelliteEarth system? (b) What is the magnitude of the gravitational force exerted by the Earth on the satellite? (c) What If? What force, if any, does the satellite exert on the Earth?

How much work is done by the Moons gravitational field on a 1 000-kg meteor as it comes in from outer space and impacts on the Moons surface?

How much energy is required to move a 1 000-kg object from the Earths surface to an altitude twice the Earths radius?

After the Sun exhausts its nuclear fuel, its ultimate fate will be to collapse to a white dwarf state. In this state, it would have approximately the same mass as it has now, but its radius would be equal to the radius of the Earth. Calculate (a) the average density of the white dwarf, (b) the surface free-fall acceleration, and (c) the gravitational potential energy associated with a 1.00-kg object at the surface of the white dwarf

An object is released from rest at an altitude h above the surface of the Earth. (a) Show that its speed at a distance r from the Earths center, where RE # r # RE 1 h, is v 5 2GME a 1 r 2 1 RE 1 h b (b) Assume the release altitude is 500 km. Perform the integral Dt 5 3 f i dt 5 23 f i dr v to find the time of fall as the object moves from the release point to the Earths surface. The negative sign appears because the object is moving opposite to the radial direction, so its speed is v 5 2dr/dt. Perform the integral numerically.

A system consists of three particles, each of mass 5.00 g, located at the corners of an equilateral triangle with sides of 30.0 cm. (a) Calculate the potential energy of the system. (b) Assume the particles are released simultaneously. Describe the subsequent motion of each. Will any collisions take place? Explain.

A space probe is fired as a projectile from the Earths surface with an initial speed of 2.00 3 104 m/s. What will its speed be when it is very far from the Earth? Ignore atmospheric friction and the rotation of the Earth.

A 500-kg satellite is in a circular orbit at an altitude of 500 km above the Earths surface. Because of air friction, the satellite eventually falls to the Earths surface, where it hits the ground with a speed of 2.00 km/s. How much energy was transformed into internal energy by means of air friction?

A treetop satellite moves in a circular orbit just above the surface of a planet, assumed to offer no air resistance. Show that its orbital speed v and the escape speed from the planet are related by the expression vesc 5 !2v.

A 1 000-kg satellite orbits the Earth at a constant altitude of 100 km. (a) How much energy must be added to the system to move the satellite into a circular orbit with altitude 200 km? What are the changes in the systems (b) kinetic energy and (c) potential energy?

A comet of mass 1.20 3 1010 kg moves in an elliptical orbit around the Sun. Its distance from the Sun ranges between 0.500 AU and 50.0 AU. (a) What is the eccentricity of its orbit? (b) What is its period? (c) At aphelion, what is the potential energy of the cometSun system? Note: 1 AU 5 one astronomical unit 5 the average distance from the Sun to the Earth 5 1.496 3 1011 m.

An asteroid is on a collision course with Earth. An astronaut lands on the rock to bury explosive charges that will blow the asteroid apart. Most of the small fragments will miss the Earth, and those that fall into the atmosphere will produce only a beautiful meteor shower. The astronaut finds that the density of the spherical asteroid is equal to the average density of the Earth. To ensure its pulverization, she incorporates into the explosives the rocket fuel and oxidizer intended for her return journey. What maximum radius can the asteroid have for her to be able to leave it entirely simply by jumping straight up? On Earth she can jump to a height of 0.500 m.

Derive an expression for the work required to move an Earth satellite of mass m from a circular orbit of radius 2RE to one of radius 3RE

(a) Determine the amount of work that must be done on a 100-kg payload to elevate it to a height of 1 000 km above the Earths surface. (b) Determine the amount of additional work that is required to put the payload into circular orbit at this elevation.

(a) What is the minimum speed, relative to the Sun, necessary for a spacecraft to escape the solar system if it starts at the Earths orbit? (b) Voyager 1 achieved a maximum speed of 125 000 km/h on its way to photograph Jupiter. Beyond what distance from the Sun is this speed sufficient to escape the solar system?

A satellite of mass 200 kg is placed into Earth orbit at a height of 200 km above the surface. (a) Assuming a circular orbit, how long does the satellite take to complete one orbit? (b) What is the satellites speed? (c) Starting from the satellite on the Earths surface, what is the minimum energy input necessary to place this satellite in orbit? Ignore air resistance but include the effect of the planets daily rotation.

A satellite of mass m, originally on the surface of the Earth, is placed into Earth orbit at an altitude h. (a) Assuming a circular orbit, how long does the satellite take to complete one orbit? (b) What is the satellites speed? (c) What is the minimum energy input necessary to place this satellite in orbit? Ignore air resistance but include the effect of the planets daily rotation. Represent the mass and radius of the Earth as ME and RE, respectively.

Ganymede is the largest of Jupiters moons. Consider a rocket on the surface of Ganymede, at the point farthest from the planet (Fig. P13.47). Model the rocket as a particle. (a) Does the presence of Ganymede make Jupiter exert a larger, smaller, or same size force on the rocket compared with the force it would exert if Ganymede were not interposed? (b) Determine the escape speed for the rocket from the planetsatellite system. The radius of Ganymede is 2.64 3 106 m, and its mass is 1.495 3 1023 kg. The distance between Jupiter and Ganymede is 1.071 3 109 m, and the mass of Jupiter is 1.90 3 1027 kg. Ignore the motion of Jupiter and Ganymede as they revolve about their center of mass.

A satellite moves around the Earth in a circular orbit of radius r. (a) What is the speed vi of the satellite? (b) Suddenly, an explosion breaks the satellite into two pieces, with masses m and 4m. Immediately after the explosion, the smaller piece of mass m is stationary with respect to the Earth and falls directly toward the Earth. What is the speed v of the larger piece immediately after the explosion? (c) Because of the increase in its speed, this larger piece now moves in a new elliptical orbit. Find its distance away from the center of the Earth when it reaches the other end of the ellipse.

At the Earths surface, a projectile is launched straight up at a speed of 10.0 km/s. To what height will it rise? Ignore air resistance.

A rocket is fired straight up through the atmosphere from the South Pole, burning out at an altitude of 250 km when traveling at 6.00 km/s. (a) What maximum distance from the Earths surface does it travel before falling back to the Earth? (b) Would its maximum distance from the surface be larger if the same rocket were fired with the same fuel load from a launch site on the equator? Why or why not?

Review. A cylindrical habitat in space 6.00 km in diameter and 30.0 km long has been proposed (by G. K. ONeill, 1974). Such a habitat would have cities, land, and lakes on the inside surface and air and clouds in the center. They would all be held in place by rotation of the cylinder about its long axis. How fast would the cylinder have to rotate to imitate the Earths gravitational field at the walls of the cylinder?

Voyager 1 and Voyager 2 surveyed the surface of Jupiters moon Io and photographed active volcanoes spewing liquid sulfur to heights of 70 km above the surface of this moon. Find the speed with which the liquid sulfur left the volcano. Ios mass is 8.9 3 1022 kg, and its radius is 1 820 km.

A satellite is in a circular orbit around the Earth at an altitude of 2.80 3 106 m. Find (a) the period of the orbit, (b) the speed of the satellite, and (c) the acceleration of the satellite.

Why is the following situation impossible? A spacecraft is launched into a circular orbit around the Earth and circles the Earth once an hour.

Let DgM represent the difference in the gravitational fields produced by the Moon at the points on the Earths surface nearest to and farthest from the Moon. Find the fraction DgM/g, where g is the Earths gravitational field. (This difference is responsible for the occurrence of the lunar tides on the Earth.)

A sleeping area for a long space voyage consists of two cabins each connected by a cable to a central hub as shown in Figure P13.56. The cabins are set spinning around the hub axis, which is connected to the rest of the spacecraft to generate artificial gravity in the cabins. A space traveler lies in a bed parallel to the outer wall as shown in Figure P13.56. (a) With r 5 10.0 m, what would the angular speed of the 60.0-kg traveler need to be if he is to experience half his normal Earth weight? (b) If the astronaut stands up perpendicular to the bed, without holding on to anything with his hands, will his head be moving at a faster, a slower, or the same tangential speed as his feet? Why? (c) Why is the action in part (b) dangerous?

(a) A space vehicle is launched vertically upward from the Earths surface with an initial speed of 8.76 km/s, which is less than the escape speed of 11.2 km/s. What maximum height does it attain? (b) A meteoroid falls toward the Earth. It is essentially at rest with respect to the Earth when it is at a height of 2.51 3 107 m above the Earths surface. With what speed does the meteorite (a meteoroid that survives to impact the Earths surface) strike the Earth?

(a) A space vehicle is launched vertically upward from the Earths surface with an initial speed of vi that is comparable to but less than the escape speed vesc. What maximum height does it attain? (b) A meteoroid falls toward the Earth. It is essentially at rest with respect to the Earth when it is at a height h above the Earths surface. With what speed does the meteorite (a meteoroid that survives to impact the Earths surface) strike the Earth? (c) What If? Assume a baseball is tossed up with an initial speed that is very small compared to the escape speed. Show that the result from part (a) is consistent with Equation 4.12.

Assume you are agile enough to run across a horizontal surface at 8.50 m/s, independently of the value of the gravitational field. What would be (a) the radius and (b) the mass of an airless spherical asteroid of uniform density 1.10 3 103 kg/m3 on which you could launch yourself into orbit by running? (c) What would be your period? (d) Would your running significantly affect the rotation of the asteroid? Explain.

Two spheres having masses M and 2M and radii R and 3R, respectively, are simultaneously released from rest when the distance between their centers is 12R. Assume the two spheres interact only with each other and we wish to find the speeds with which they collide. (a) What two isolated system models are appropriate for this system? (b) Write an equation from one of the models and solve it for vS 1, the velocity of the sphere of mass M at any time after release in terms of vS 2, the velocity of 2M. (c) Write an equation from the other model and solve it for speed v1 in terms of speed v2 when the spheres collide. (d) Combine the two equations to find the two speeds v1 and v2 when the spheres collide.

Two hypothetical planets of masses m1 and m2 and radii r1 and r2, respectively, are nearly at rest when they are an infinite distance apart. Because of their gravitational attraction, they head toward each other on a collision course. (a) When their center-to-center separation is d, find expressions for the speed of each planet and for their relative speed. (b) Find the kinetic energy of each planet just before they collide, taking m1 5 2.00 3 1024 kg, m2 5 8.00 3 1024 kg, r1 5 3.00 3 106 m, and r2 5 5.00 3 106 m. Note: Both the energy and momentum of the isolated two-planet system are constant.

(a) Show that the rate of change of the free-fall acceleration with vertical position near the Earths surface is dg dr 5 2 2GME RE 3 This rate of change with position is called a gradient. (b) Assuming h is small in comparison to the radius of the Earth, show that the difference in free-fall acceleration between two points separated by vertical distance h is 0 Dg 0 5 2GMEh RE 3 (c) Evaluate this difference for h 5 6.00 m, a typical height for a two-story building.

A ring of matter is a familiar structure in planetary and stellar astronomy. Examples include Saturns rings and a ring nebula. Consider a uniform ring of mass 2.36 3 1020 kg and radius 1.00 3 108 m. An object of mass 1 000 kg is placed at a point A on the axis of the ring, 2.00 3 108 m from the center of the ring (Fig. P13.63). When the object is released, the attraction of the ring makes the object move along the axis toward the center of the ring (point B). (a) Calculate the gravitational potential energy of the objectring system when the object is at A. (b) Calculate the gravitational potential energy of the system when the object is at B. (c) Calculate the speed of the object as it passes through B.

A spacecraft of mass 1.00 3 104 kg is in a circular orbit at an altitude of 500 km above the Earths surface. Mission Control wants to fire the engines in a direction tangent to the orbit so as to put the spacecraft in an elliptical orbit around the Earth with an apogee of 2.00 3 104 km, measured from the Earths center. How much energy must be used from the fuel to achieve this orbit? (Assume that all the fuel energy goes into increasing the orbital energy. This model will give a lower limit to the required energy because some of the energy from the fuel will appear as internal energy in the hot exhaust gases and engine parts.)

Review. As an astronaut, you observe a small planet to be spherical. After landing on the planet, you set off, walking always straight ahead, and find yourself returning to your spacecraft from the opposite side after completing a lap of 25.0 km. You hold a hammer and a falcon feather at a height of 1.40 m, release them, and observe that they fall together to the surface in 29.2 s. Determine the mass of the planet.

A certain quaternary star system consists of three stars, each of mass m, moving in the same circular orbit of radius r about a central star of mass M. The stars orbit in the same sense and are positioned one-third of a revolution apart from one another. Show that the period of each of the three stars is given by T 5 2p r 3 G1M 1 m/"32

Studies of the relationship of the Sun to our galaxy the Milky Wayhave revealed that the Sun is located near the outer edge of the galactic disc, about 30 000 ly (1 ly 5 9.46 3 1015 m) from the center. The Sun has an orbital speed of approximately 250 km/s around the galactic center. (a) What is the period of the Suns galactic motion? (b) What is the order of magnitude of the mass of the Milky Way galaxy? (c) Suppose the galaxy is made mostly of stars of which the Sun is typical. What is the order of magnitude of the number of stars in the Milky Way?

Review. Two identical hard spheres, each of mass m and radius r, are released from rest in otherwise empty space with their centers separated by the distance R. They are allowed to collide under the influence of their gravitational attraction. (a) Show that the magnitude of the impulse received by each sphere before they make contact is given by [Gm3(1/2r 2 1/R)]1/2. (b) What If? Find the magnitude of the impulse each receives during their contact if they collide elastically

The maximum distance from the Earth to the Sun (at aphelion) is 1.521 3 1011 m, and the distance of closest approach (at perihelion) is 1.471 3 1011 m. The Earths orbital speed at perihelion is 3.027 3 104 m/s. Determine (a) the Earths orbital speed at aphelion and the kinetic and potential energies of the EarthSun system (b) at perihelion and (c) at aphelion. (d) Is the total energy of the system constant? Explain. Ignore the effect of the Moon and other planets.

Many people assume air resistance acting on a moving object will always make the object slow down. It can, however, actually be responsible for making the object speed up. Consider a 100-kg Earth satellite in a circular orbit at an altitude of 200 km. A small force of air resistance makes the satellite drop into a circular orbit with an altitude of 100 km. (a) Calculate the satellites initial speed. (b) Calculate its final speed in this process. (c) Calculate the initial energy of the satelliteEarth system. (d) Calculate the final energy of the system. (e) Show that the system has lost mechanical energy and find the amount of the loss due to friction. (f) What force makes the satellites speed increase? Hint: You will find a free-body diagram useful in explaining your answer.

X-ray pulses from Cygnus X-1, the first black hole to be identified and a celestial x-ray source, have been recorded during high-altitude rocket flights. The signals can be interpreted as originating when a blob of ionized matter orbits a black hole with a period of 5.0 ms. If the blob is in a circular orbit about a black hole whose mass is 20MSun, what is the orbit radius?

Show that the minimum period for a satellite in orbit around a spherical planet of uniform density r is Tmin 5 3p Gr independent of the planets radius.

Astronomers detect a distant meteoroid moving along a straight line that, if extended, would pass at a distance 3RE from the center of the Earth, where RE is the Earths radius. What minimum speed must the meteoroid have if it is not to collide with the Earth?

Two stars of masses M and m, separated by a distance d, revolve in circular orbits about their center of mass (Fig. P13.74). Show that each star has a period given by T 2 5 4p2d3 G 1M 1 m2

Two identical particles, each of mass 1 000 kg, are coasting in free space along the same path, one in front of the other by 20.0 m. At the instant their separation distance has this value, each particle has precisely the same velocity of 800 i^ m/s. What are their precise velocities when they are 2.00 m apart?

Consider an object of mass m, not necessarily small compared with the mass of the Earth, released at a distance of 1.20 3 107 m from the center of the Earth. Assume the Earth and the object behave as a pair of particles, isolated from the rest of the Universe. (a) Find the magnitude of the acceleration arel with which each starts to move relative to the other as a function of m. Evaluate the acceleration (b) for m 5 5.00 kg, (c) for m 5 2 000 kg, and (d) for m 5 2.00 3 1024 kg. (e) Describe the pattern of variation of arel with m.

As thermonuclear fusion proceeds in its core, the Sun loses mass at a rate of 3.64 3 109 kg/s. During the 5 000-yr period of recorded history, by how much has the length of the year changed due to the loss of mass from the Sun? Suggestions: Assume the Earths orbit is circular. No external torque acts on the EarthSun system, so the angular momentum of the Earth is constant.

The Solar and Heliospheric Observatory (SOHO) spacecraft has a special orbit, located between the Earth and the Sun along the line joining them, and it is always close enough to the Earth to transmit data easily. Both objects exert gravitational forces on the observatory. It moves around the Sun in a near-circular orbit that is smaller than the Earths circular orbit. Its period, however, is not less than 1 yr but just equal to 1 yr. Show that its distance from the Earth must be 1.48 3 109 m. In 1772, Joseph Louis Lagrange determined theoretically the special location allowing this orbit. Suggestions: Use data that are precise to four digits. The mass of the Earth is 5.974 3 1024 kg. You will not be able to easily solve the equation you generate; instead, use a computer to verify that 1.48 3 109 m is the correct value.

The oldest artificial satellite still in orbit is Vanguard I, launched March 3, 1958. Its mass is 1.60 kg. Neglecting atmospheric drag, the satellite would still be in its initial orbit, with a minimum distance from the center of the Earth of 7.02 Mm and a speed at this perigee point of 8.23 km/s. For this orbit, find (a) the total energy of the satelliteEarth system and (b) the magnitude of the angular momentum of the satellite. (c) At apogee, find the satellites speed and its distance from the center of the Earth. (d) Find the semimajor axis of its orbit. (e) Determine its period.

A spacecraft is approaching Mars after a long trip from the Earth. Its velocity is such that it is traveling along a parabolic trajectory under the influence of the gravitational force from Mars. The distance of closest approach will be 300 km above the Martian surface. At this point of closest approach, the engines will be fired to slow down the spacecraft and place it in a circular orbit 300 km above the surface. (a) By what percentage must the speed of the spacecraft be reduced to achieve the desired orbit? (b) How would the answer to part (a) change if the distance of closest approach and the desired circular orbit altitude were 600 km instead of 300 km? (Note: The energy of the spacecraftMars system for a parabolic orbit is E 5 0.)