Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 6: Problem 733

The escape velocity from the earth is about \(11 \mathrm{kms}^{-1}\). The escape velocity from a planet having twice the radius and the same mean density as the earth is \(=\ldots \ldots \ldots \ldots \mathrm{kms}^{-1}\). (A) 22 (B) 11 (C) \(5.5\) (D) \(15.5\)

Short Answer

Expert verified
The escape velocity of the new planet is 22 km/s.

Step by step solution

01

Recall the escape velocity formula

The formula for escape velocity is given by: \(v_e = \sqrt{\frac{2GM}{r}}\), where \(v_e\) is the escape velocity, \(G\) is the gravitational constant, \(M\) is the mass of the planet, and \(r\) is the radius of the planet.
02

Find the new planet's mass

The volume of a sphere is given by the formula \(V = \frac{4}{3}\pi r^3\). Since the new planet has twice the radius of Earth, its volume will be 8 times larger. Since the new planet has the same mean density as Earth, its mass will also be 8 times larger. Let the mass of Earth be \(M_e\), the mass of the new planet is \(8M_e\).
03

Find the new planet's radius

The new planet's radius is twice the radius of Earth. Let the radius of Earth be \(r_e\), the radius of the new planet is \(2r_e\).
04

Calculate the escape velocity of the new planet

Using the escape velocity formula, substitute the mass and radius of the new planet to find its escape velocity: \(v_{e_{new}} = \sqrt{\frac{2G(8M_e)}{2r_e}}\) Notice that the \(2\) in the numerator and the \(2\) in the denominator will cancel out: \(v_{e_{new}} = \sqrt{\frac{8GM_e}{r_e}}\)
05

Recall the escape velocity of Earth and make the comparison

We know that the escape velocity from Earth is about 11 km/s, so: \(v_{e_{Earth}} = \sqrt{\frac{2GM_e}{r_e}} = 11\) Now, we can square both sides of this equation and obtain: \(\frac{2GM_e}{r_e} = 121\) Now, compare the escape velocity of the new planet with that of Earth: \(\frac{v_{e_{new}}}{v_{e_{Earth}}} = \frac{\sqrt{\frac{8GM_e}{r_e}}}{\sqrt{\frac{2GM_e}{r_e}}} = \sqrt{\frac{8GM_e}{r_e} \cdot \frac{r_e}{2GM_e}} = \sqrt{4} = 2\) This implies that the escape velocity of the new planet is twice that of Earth, or: \(v_{e_{new}} = 2 \cdot 11 = 22 \mathrm{kms}^{-1}\) So the correct answer is (A) 22.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

4 A satellite which is geostationary in a particular orbit is taken to another orbit. Its distance from the center of earth in new orbit is two times of the earlier orbit. The time period in second orbit is $\ldots \ldots \ldots \ldots$ hours. (A) \(4.8\) (B) \(48 \sqrt{2}\) (C) 24 (D) \(24 \sqrt{2}\)

A small satellite is revolving near earth's surface. Its orbital velocity will be nearly \(=\ldots \ldots \ldots \ldots \mathrm{kms}^{-1}\). (A) 8 (B) 4 (C) 6 (D) \(11.2\)

An artificial satellite moving in a circular orbit around earth has a total (kinetic + potential energy) \(E_{0}\), its potential energy is (A) \(-\mathrm{E}_{0}\) (B) \(1.5 \mathrm{E}_{0}\) (C) \(2 \mathrm{E}_{0}\) (D) \(\mathrm{E}_{0}\)

Two satellites \(\mathrm{A}\) and \(\mathrm{B}\) go round a planet in circular orbits having radii \(4 \mathrm{R}\) and \(\mathrm{R}\) respectively If the speed of satellite \(\mathrm{A}\) is \(3 \mathrm{v}\), then speed of satellite \(\mathrm{B}\) is (A) \((3 \mathrm{v} / 2)\) (B) \((4 \mathrm{v} / 2)\) (C) \(6 \mathrm{v}\) (D) \(12 \mathrm{v}\)

Suppose the gravitational force varies inversely as the nth power of distance the time period of planet in circular orbit of radius \(\mathrm{R}\) around the sun will be proportional to (A) \(\mathrm{R}^{[(\mathrm{n}+1) / 2]}\) (B) \(\mathrm{R}^{[(\mathrm{n}-1) / 2]}\) (C) \(\mathrm{R}^{\mathrm{n}}\) (D) \(\mathrm{R}^{[(\mathrm{n}-1) / 2]}\)
See all solutions

Recommended explanations on English Textbooks

Summary Text

Read Explanation

Key Concepts in Language and Linguistics

Read Explanation

Essay Plan

Read Explanation

Free Response Essay

Read Explanation

Text Comparison

Read Explanation

Morphology

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free