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Chapter 13: Q. 33 (page 354)

In $2000$ , NASA placed a satellite in orbit around an asteroid. Consider a spherical asteroid with a mass of $1 脳 10^{16} k g$ and a radius of $8.8 k m .$
a. What is the speed of a satellite orbiting $5 k m$ above the surface?
b. What is the escape speed from the asteroid?

Short Answer

Expert verified

a. The speed of the satellite is $6.95 m / s .$

b. The escape speed from the asteroid is $12.31 m / s .$

Step by step solution

01

Given information a

Height = $5000 m,$ mass of asteroid (M) = $1.0 脳 10^{16} kg$ , Radius of asteroid (R) = $8800 m .$

02

Calculation a

For a satellite to revolve around the planet the necessary centripetal force is provided by the gravitational force of attraction between the planet and the satellite.

$F_{c} = centripetal force = \frac{m v^{2}}{r}$

Where orbital radius of satellite r = R + h

And $F_{g} = \frac{G M m}{r^{2}}$

Now $F_{c} = F_{g} \frac{m v^{2}}{r} = \frac{G M m}{r^{2}} v = \sqrt{\frac{G M}{r}} = \sqrt{\frac{G M}{R + h}}$

Substituting values and solving we get : $v = 6.95 m / s$ .

03

Final answer a

The speed of the satellite is $6.95 m / s .$

04

Given information b

Mass of the asteroid (M) = $1.0 脳 10^{16} kg,$ radius of asteroid = $8800 m .$

05

Calculation b

The escape speed from a planet is given by the formula : $v_{e} = \sqrt{\frac{2 G M}{R}}$

Substituting values in the formula we get : $v_{e} = 12.31 m / s .$

06

Final answer b

The escape speed from the planet is $12.31 m / s .$

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Most popular questions from this chapter

The mass of Jupiter is 300 times the mass of the earth. Jupiter orbits the sun with TJupiter = 11.9 yr in an orbit with r_Jupiter = 5.2r_earth. Suppose the earth could be moved to the distance
of Jupiter and placed in a circular orbit around the sun. Which of the following describes the earth鈥檚 new period?
Explain.
a. 1 yr
b. Between 1 yr and 11.9 yr
c. 11.9 yr
d. More than 11.9 yr
e. It would depend on the earth鈥檚 speed.
f. It鈥檚 impossible for a planet of earth鈥檚 mass to orbit at the
distance of Jupiter.

Let鈥檚 look in more detail at how a satellite is moved from one circular orbit to another. FIGURE $CP13.71$ shows two circular orbits, of radii localid="1651418485730" $r_{1}$ and localid="1651418489556" $r_{2}$ , and an elliptical orbit that connects them. Points $1$ and $2$ 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" $1$ and localid="1651418499267" $2$ are localid="1651418492993" $v_{1}^{鈥�} = \sqrt{\frac{2 G M (r_{2} / r_{1})}{r_{1} + r_{2}}}$ and localid="1651418509687" $v_{2}^{鈥�} = \sqrt{\frac{2 G M (r_{1} / r_{2})}{r_{1} + r_{2}}}$ The prime indicates that these are the velocities on the elliptical orbit. Both reduce to Equation $13.22$ if localid="1651418513535" $r_{1} = r_{2 = r}$ .

b. Consider a localid="1651418519576" $1000 k g$ communications satellite that needs to be boosted from an orbit localid="1651418573632" $300 k m$ above the earth to a geosynchronous orbit localid="1651418578672" $35, 900 k m$ above the earth. Find the velocity localid="1651418584351" $v_{1}$ on the inner circular orbit and the velocity localid="1651418590277" $v = 1$ 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" $v = 2$ 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" $13.6$ .

A space station orbits the sun at the same distance as the earth but on the opposite side of the sun. A small probe is fired away from the station. What minimum speed does the probe need to escape the solar system?

Why is the gravitational potential energy of two masses negative? Note that saying 鈥渂ecause that鈥檚 what the equation gives鈥� is not an explanation.

What is the ratio of the sun鈥檚 gravitational force on you to the
earth鈥檚 gravitational force on you?

See all solutions

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