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 bythe expression Vesc√2v. (Submit a file with a maximum size of 1 MB.)Choose File No file chosen

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Answered: A "treetop satellite" moves in a…

Classical Dynamics of Particles and Systems

5th Edition

ISBN:9780534408961

Author:Stephen T. Thornton, Jerry B. Marion

Publisher:Stephen T. Thornton, Jerry B. Marion

Chapter8: Central-force Motion

Section: Chapter Questions

Problem 8.3P

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Similar questions

What is the orbital radius of an Earth satellite having a period of 1.00 h? (b) What is unreasonable about this result?
Show that the areal velocity for a circular orbit of radius r about a mass M is At=12GMr . Does your expression give the correct value for Earth’s areal vilocity about the Sun?
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The “mean” orbital radius listed for astronomical objects orbiting the Sun is typically not an integrated average but is calculated such that it gives the correct period when applied to the equation for circular orbits. Given that, what is the mean orbital radius in terms of aphelion and perihelion?
Using the technique shown in Satellite Orbits and Energy, show that two masses m1 and m2 in circular orbits about their common center of mass, will have total energy E=K+E=K1+K2Gm1m2r=Gm1m22r . We have shown the kinetic energy of both masses explicitly. (Hint: The masses orbit at radii r1 and r2 , respectively, where r=r1+r2 . Be sure not to confuse the radius needed for centripetal acceleration with that for the gravitational force.)
In a later chapter, you will find that the weight of a particle varies with altitude such that w=mgr02r2where r0is the radius of Earth and ris the distance from Earth’s center. If the particle is fired vertically with velocity v0from Earth’s surface, determine its velocity as a function of position r. (Hint: use adr=vdv, the rearrangement mentioned in the text.)
On a planet whose radius is 1.2107m , the acceleration due to gravity is 18m/s2 . What is the mass of the planet?
Check Your Understanding The nearly circular orbit of Saturn has an average radius of about 9.5 AU and has a period of 30 years, whereas Uranus averages about 19 AU and has a period of 84 years. Is this consistent with our results for Halley’s comet?
2. (a) What are meant by gravitational potential and gravitational field? How far away fromthe earth does the acceleration due to gravity become 10% of its value on the earth’ssurface? NB:The radius of the earth is 6370 km. (b) What is meant by escape velocity? Show that the escape velocity from the surface of earthis √2 times the velocity of projection of an artificial satellite orbiting close around the earth.Obtain the escape velocity of an atmospheric particle 100 km above the earth’s surface. Giventhat ME = 5.98 × 1024 kg, RE = 6370 km, G = 6.67 × 10-11 N.m2/kg2.
(a) Calculate the escape velocity of an object on the Earth, assuming a mean gravitational acceleration of 9.81 m s-! and mean Earth radius of 6371 km. (b) What is the escape velocity of the same object on the Moon?
A mood of a mass m and raduis a is orbiting a planet of mass M and if radius b at a distance d (center to center) in a circular orbit. Derive an expression for the speed of the mood v in terms of M,d and the gravitational constant G.
At the surface of planet X, a 1 kg object weighs 4 N (planet radius R=106m). A space probe passes by planet X with the nearest point (A) at 8R from the center. When the probe was very far away it had a speed of sqrt(2gxR), where gx is the acceleration of gravity at the planet surface. Find a value for the speed of this probe when it is at point A, vA.

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