/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 78 The mass of a spaceship is \(100... [FREE SOLUTION] | 91影视

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Chapter 7: Problem 78

The mass of a spaceship is \(1000 \mathrm{~kg}\). It is to be launched from the earth's surface out into free space. The value of \(g\) and \(r\) (radius of earth) are \(10 \mathrm{~m} / \mathrm{s}^{2}\) and \(6400 \mathrm{~km}\) respectively. The required energy for this work will be (A) \(6.4 \times 10^{11} \mathrm{~J}\) (B) \(6.4 \times 10^{8} \mathrm{~J}\) (C) \(6.4 \times 10^{9} \mathrm{~J}\) (D) \(6.4 \times 10^{10} \mathrm{~J}\)

Short Answer

Expert verified
The short answer is: The required energy for this work is \(6.4 \times 10^{11} \mathrm{~J}\) (Option A).

Step by step solution

01

Identify the known values

We know the following values: Mass of spaceship (m) = 1000 kg Gravitational acceleration (g) = 10 m/s虏 Radius of Earth (r) = 6400 km = 6400 脳 10^3 m
02

Calculate the gravitational potential energy on Earth's surface

We will use the formula for gravitational potential energy (PE): PE = m 脳 g 脳 h In this case, h (height) is equal to the radius of the Earth. Therefore, we have: PE = 1000 kg 脳 10 m/s虏 脳 6400 脳 10^3 m PE = 6.4 脳 10^7 kg 脳 m虏/s虏 or Joules (J)
03

Calculate the energy required to move the spaceship to infinity

We will use the formula for gravitational potential energy at infinity: PE_infinity = - G 脳 (m 脳 M) / r Where G is the gravitational constant (approximately 6.674 脳 10^-11 Nm虏/kg虏), M is the mass of Earth (approximately 5.972 脳 10^24 kg), and r is the radius of Earth (6400 脳 10^3 m). PE_infinity = - (6.674 脳 10^-11 Nm虏/kg虏) 脳 (1000 kg 脳 5.972 脳 10^24 kg) / (6400 脳 10^3 m) PE_infinity 鈮� - 6.3934 脳 10^11 J
04

Calculate the total energy required to launch the spaceship

The total energy required to launch the spaceship is the difference between the gravitational potential energy on Earth's surface and the energy needed to move it to infinity: Energy_required = PE_infinity - PE Energy_required 鈮� - 6.3934 脳 10^11 J - 6.4 脳 10^7 J Energy_required 鈮� 6.4 脳 10^11 J Comparing this energy value with the given options, we find that the answer is: (A) 6.4 脳 10^11 J

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Most popular questions from this chapter

If three uniform spheres, each having mass \(M\) and radius \(R\), are kept in such a way that each touches the other two, the magnitude of the gravitational force on any sphere due to the other two is (A) \(\frac{G M^{2}}{4 R^{2}}\) (B) \(\frac{2 G M^{2}}{R^{2}}\) (C) \(\frac{2 G M^{2}}{4 R^{2}}\) (D) \(\frac{\sqrt{3} G M^{2}}{4 R^{2}}\)

The change in the value of \(g\) at a height \(h\) above the surface of the earth is the same as that of a depth \(d\) below the surface of earth. When both \(d\) and \(h\) are much smaller than the radius of earth, then which one of the following is correct? (A) \(d=\frac{h}{2}\) (B) \(d=\frac{3 h}{2}\) (C) \(d=2 h\) (D) \(d=h\)

Four particles, each of mass \(M\) and equidistant from each other, move along a circle of radius \(R\) under the action of their mutual gravitational attraction. The speed of each particle is (A) \(\sqrt{\frac{G M}{R}}\) (B) \(\sqrt{2 \sqrt{2} \frac{G M}{R}}\) (C) \(\sqrt{\frac{G M}{R}(1+2 \sqrt{2})}\) (D) \(\frac{1}{2} \sqrt{\frac{G M}{R}(1+2 \sqrt{2})}\)

Three equal masses \(m \mathrm{~kg}\) are placed at the vertices of an equilateral triangle of side \(a\) metre. The gravitational potential energy equals to (A) \(-\frac{3 G m^{2}}{a}\) (B) \(-\frac{3 G m}{a^{2}}\) (C) \(-\frac{3 G m}{a}\) (D) \(\frac{3 G m^{2}}{a}\)

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