Q. 49 A starship is circling a distant... [FREE SOLUTION]

Chapter 13: Q. 49 (page 355)

A starship is circling a distant planet of radius R. The astronauts find that the free-fall acceleration at their altitude is half the value at the planet鈥檚 surface. How far above the surface are they orbiting? Your answer will be a multiple of $R$ .

Short Answer

Expert verified

The starship is orbiting above $0.414 R$ above the surface.

Step by step solution

Step 1. Given Information.

Acceleration on the surface of a planet = 2 times of the acceleration at a distance from the surface of a planet $- - - - (1)$

Step 1. Apply the formula.

Acceleration on the surface of a planet with radius $R = \frac{G M}{R^{2}} - - - - (i)$

Acceleration at a distance $h$ from the surface of a planet with radius $R = \frac{G M}{{(R + h)}^{2}} - - - - (i i)$

Step 3. Substitute the formulas.Substitute the equation i and ii into 1 and obtain, GMR2=2·GMR+h2.Then solve the obtained equation.

Substitute the equation $(i)$ and $(i i)$ into $(1)$ and obtain, $\frac{G M}{R^{2}} = 2 路 \frac{G M}{{(R + h)}^{2}}$ .

Then solve the obtained equation.

2 R^{2} = {(R + h)}^{2} \frac{R + h}{R} = \sqrt{2} 1 + \frac{h}{R} = \sqrt{2} \frac{h}{R} = \sqrt{2} - 1 \frac{h}{R} = 1.414 - 1 h = 0.414 R

Step 4. Final Answer.

The starship is orbiting $0.414 R$ above the surface.

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A starship is circling a distant planet of radius R. The astronauts find that the free-fall acceleration at their altitude is half the value at the planet鈥檚 surface. How far above the surface are they orbiting? Your answer will be a multiple of $R$ .

Short Answer

Step by step solution

Step 1. Given Information.

Step 1. Apply the formula.

Step 3. Substitute the formulas.Substitute the equation i and ii into 1 and obtain, GMR2=2·GMR+h2.Then solve the obtained equation.

Step 4. Final Answer.

One App. One Place for Learning.

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91影视

A starship is circling a distant planet of radius R. The astronauts find that the free-fall acceleration at their altitude is half the value at the planet鈥檚 surface. How far above the surface are they orbiting? Your answer will be a multiple of R.

Short Answer

Step by step solution

Step 1. Given Information.

Step 1. Apply the formula.

Step 3. Substitute the formulas.Substitute the equation i and ii into 1 and obtain, GMR2=2·GMR+h2.Then solve the obtained equation.

Step 4. Final Answer.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Astrophysics

Medical Physics

Measurements

Translational Dynamics

Further Mechanics and Thermal Physics

Space Physics

Study anywhere. Anytime. Across all devices.

A starship is circling a distant planet of radius R. The astronauts find that the free-fall acceleration at their altitude is half the value at the planet鈥檚 surface. How far above the surface are they orbiting? Your answer will be a multiple of $R$ .