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Chapter 8: Problem 32

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

Short Answer

Expert verified
The radius to which Earth would have to shrink for escape speed at its surface to be 30 km/s, with its mass unchanged, is approximately \(1.18 \times 10^{7} \, m\).

Step by step solution

01

Identify Given Variables

The given variables include the escape speed (\(v\)) which is 30 km/s and needs to be converted to m/s equal to \(30 \times 1000 = 30000 \, m/s\), the gravitational constant (\(G\)) which is \(6.67 \times 10^{-11} \, N(m/kg)^2\), and the mass of the Earth (\(M\)) which is \(5.97 \times 10^{24} \, kg\). The radius (\(r\)) of the Earth is what we will solve for.
02

Apply the Escape Speed Formula

The escape speed formula \(v = \sqrt{(2GM / r)}\) is used to solve for the radius of the Earth. We know all the variables except for \(r\), which we will rearrange the formula to solve for.
03

Rearrange the Formula to Solve for 'r'

By rearranging the escape speed formula, we get \(r = 2GM / v^2\). By substituting the known values, we can solve for \(r\).
04

Perform the Calculation

Substitute the known values into the formula \(r = \frac{2 \times 6.67 \times 10^{-11} \, N(m/kg)^2 \times 5.97 \times 10^{24} \, kg}{(30000 \, m/s)^2}\) to get \(r \approx 1.18 \times 10^{7}m\)

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Most popular questions from this chapter

Given Earth's mass, the Moon's distance and orbital period, and the value of \(G,\) could you calculate the Moon's mass? If yes, how? If no, why not?

A projectile is launched vertically upward from a planet of mass \(M\) and radius \(R ;\) its initial speed is twice the escape speed. Derive an expression for its speed as a function of the distance \(r\) from the planet's center.

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.

Show that the form \(\Delta U=m g \Delta r\) follows from Equation 8.5 when \(r_{1} \simeq r_{2} .\) [Hint: Write \(r_{2}=r_{1}+\Delta r\) and apply the binomial approximation (Appendix A).]

To what fraction of its current radius would Earth have to shrink (with no change in mass) for the gravitational acceleration at its surface to triple?
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