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Chapter 13: Q. 34 (page 354)
Pluto moves in a fairly elliptical orbit around the sun. Pluto鈥檚 speed at its closest approach of km is 6.12 km/s. What is Pluto鈥檚 speed at the most distant point in its orbit, where it iskm from the sun?
The Pluto's speed at the most distant point in its orbit is.
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A proposed space elevator would consist of a cable stretching from the earth鈥檚 surface to a satellite, orbiting far in space, that would keep the cable taut. A motorized climber could slowly carry rockets to the top, where they could be launched away from the earth using much less energy. What would be the escape speed for a craft launched from a space elevator at a height of ? Ignore the earth鈥檚 rotation.
In 2014, the European Space Agency placed a satellite in orbit around comet 67P/Churyumov-Gerasimenko and then landed a probe on the surface. The actual orbit was elliptical, but we鈥檒l approximate it as a 50-km-diameter circular orbit with a period of 11 days.
a. What was the satellite鈥檚 orbital speed around the comet? (Both the comet and the satellite are orbiting the sun at a much higher speed.)
b. What is the mass of the comet?
c. The lander was pushed from the satellite, toward the comet, at a speed of 70 cm/s, and it then fell鈥攖aking about 7 hours鈥攖o the surface. What was its landing speed? The comet鈥檚 shape is
irregular, but on average it has a diameter of 3.6 km.
Two Jupiter-size planets are released from rest 1.0 x 1011m apart. What are their speeds as they crash together?
What is the escape speed from Jupiter?
Let鈥檚 look in more detail at how a satellite is moved from one circular orbit to another. FIGURE shows two circular orbits, of radii localid="1651418485730" and localid="1651418489556" , and an elliptical orbit that connects them. Points and are at the ends of the semimajor axis of the ellipse.
a. A satellite moving along the elliptical orbit has to satisfy two conservation laws. Use these two laws to prove that the velocities at points localid="1651418503699" and localid="1651418499267" are localid="1651418492993" and localid="1651418509687" The prime indicates that these are the velocities on the elliptical orbit. Both reduce to Equation if localid="1651418513535" .
b. Consider a localid="1651418519576" communications satellite that needs to be boosted from an orbit localid="1651418573632" above the earth to a geosynchronous orbit localid="1651418578672" above the earth. Find the velocity localid="1651418584351" on the inner circular orbit and the velocity localid="1651418590277" at the low point on the elliptical orbit that spans the two circular orbits.
c. How much work must the rocket motor do to transfer the satellite from the circular orbit to the elliptical orbit?
d. Now find the velocity localid="1651418596735" at the high point of the elliptical orbit and the velocity v2 of the outer circular orbit.
e. How much work must the rocket motor do to transfer the satellite from the elliptical orbit to the outer circular orbit?
f. Compute the total work done and compare your answer to the result of Example localid="1651418602767" .
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