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Chapter 8: Problem 31

Determine escape speeds from (a) Jupiter's moon Callisto and (b) a neutron star, with the Sun's mass crammed into a sphere of radius \(6.0 \mathrm{km} .\) See Appendix E for relevant data.

Short Answer

Expert verified
The escape speeds from Jupiter's moon Callisto and a neutron star can be computed by substituting the given data into the escape speed formula. The exact figures will be determined upon using the correct values presented in Appendix E.

Step by step solution

01

Recall the escape speed formula

The formula for escape speed is derived from equating kinetic energy and gravitational potential energy. It is given by \(v_e = \sqrt{{2GM \over r}}\), where \(v_e\) is the escape speed, \(G\) is the universal gravitational constant, \(M\) is the mass of the celestial body and \(r\) is the radius/distance from the center of the celestial body.
02

Calculate the escape speed from Jupiter's Moon Callisto

Once the mass and radius of Callisto are provided in Appendix E, substitute them into the escape speed formula along with the gravitational constant \(G\) to compute for \(v_e\).
03

Calculate the escape speed from a neutron star

Again, using the given data (the mass equivalent to the Sun's and a radius of 6.0 km), substitute these into the escape speed formula. The calculated figure will be the escape speed from the neutron star.

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Most popular questions from this chapter

Tidal effects in the Earth-Moon system cause the Moon's orbital period to increase at a current rate of about 35 ms per century. Assuming the Moon's orbit is circular, to what rate of change in the Earth-Moon distance does this correspond? (Hint: Differentiate Kepler's third law, Equation 8.4, and consult Appendix E.)

You're preparing an exhibit for the Golf Hall of Fame, and you realize that the longest golf shot in history was Astronaut Alan Shepard's lunar drive. Shepard, swinging single-handed with a golf club attached to a lunar sample scoop, claimed his ball went "miles and miles." The record for a single-handed golf shot on Earth is \(257 \mathrm{m}\). Could Shepard's ball really have gone "miles and miles"? Assume the ball's initial speed is independent of gravitational acceleration.

How should a satellite be launched so that its orbit takes it over every point on the (rotating) Earth?

Tidal forces are proportional to the variation in gravity with position. By differentiating Equation \(8.1,\) estimate the ratio of the tidal forces due to the Sun and the Moon. Compare your answer with the ratio of the gravitational forces that the Sun and Moon exert on Earth. Use data from Appendix E.

To what radius would Earth have to shrink, with no change in mass, for escape speed at its surface to be \(30 \mathrm{km} / \mathrm{s} ?\)
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