Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 12: Problem 21

Satellites in low orbit around the Earth lose energy from colliding with the gases of the upper atmosphere, causing them to slowly spiral inward. What happens to their kinetic energy as they fall inward?

Short Answer

Expert verified
Answer: As the satellite loses energy and spirals inward, its kinetic energy increases to maintain the conservation of mechanical energy. The satellite gains tangential velocity as it moves into an orbit closer to the Earth, resulting in the increase in kinetic energy.

Step by step solution

01

Understand gravitational potential energy

Gravitational potential energy (U) is the energy an object possesses due to its position relative to a massive body, such as Earth. It is given by the formula: U = -G * m * M / r where G is the gravitational constant, m is the mass of the satellite, M is the mass of the Earth, and r is the distance between the center of the Earth and the satellite.
02

Understand kinetic energy

Kinetic energy (K) is the energy of an object due to its motion. It can be calculated using the formula: K = (1/2) * m * v^2 where m is the mass of the satellite, and v is its velocity.
03

Understand conservation of mechanical energy

The total mechanical energy (E) of a system is the sum of its kinetic energy (K) and potential energy (U) - that is, E = K + U. According to the conservation of mechanical energy principle, the total mechanical energy of a system remains constant if no external force acts upon it. In this case, the external force is the collisions with the atmosphere. We'll need to calculate the changes in total mechanical energy due to these collisions.
04

Calculate the change in gravitational potential energy

As the satellite spirals inward, its distance (r) from the Earth's center decreases. This means that the gravitational potential energy of the satellite (U) becomes less negative, or in other words, it increases. Since the mass of the satellite and Earth and the gravitational constant do not change, the change in U is directly related to the change in r.
05

Calculate the change in kinetic energy

In order for the total mechanical energy to be conserved, the increase in gravitational potential energy must be matched by an increase in the satellite's kinetic energy. To maintain a stable orbit, as the satellite moves inward, its tangential velocity must increase. Using the formula K = (1/2) * m * v^2, the increase in kinetic energy can be calculated.
06

Summarize the changes in the satellite's kinetic energy

As the satellite loses energy due to collisions with the Earth's upper atmosphere, it spirals inward, leading to an increase in its gravitational potential energy. To conserve mechanical energy, the satellite's kinetic energy also increases, as it gains tangential velocity while moving into an orbit closer to the Earth. Thus, the satellite's kinetic energy increases as it falls inward.

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

Is the orbital speed of the Earth when it is closest to the Sun greater than, less than, or equal to the orbital speed when it is farthest from the Sun? Explain.

Even though the Moon does not have an atmosphere, the trajectory of a projectile near its surface is only approximately a parabola. This is because the acceleration due to gravity near the surface of the Moon is only approximately constant. Describe as precisely as you can the actual shape of a projectile's path on the Moon, even one that travels a long distance over the surface of the Moon.

Imagine that two tunnels are bored completely through the Earth, passing through the center. Tunnel 1 is along the Earth's axis of rotation, and tunnel 2 is in the equatorial plane, with both ends at the Equator. Two identical balls, each with a mass of \(5.00 \mathrm{~kg}\), are simultaneously dropped into both tunnels. Neglect air resistance and friction from the tunnel walls. Do the balls reach the center of the Earth (point \(C\) ) at the same time? If not, which ball reaches the center of the Earth first?

For two identical satellites in circular motion around the Earth, which statement is true? a) The one in the lower orbit has less total energy. b) The one in the higher orbit has more kinetic energy. c) The one in the lower orbit has more total energy. d) Both have the same total energy.

Standing on the surface of a small spherical moon whose radius is \(6.30 \cdot 10^{4} \mathrm{~m}\) and whose mass is \(8.00 \cdot 10^{18} \mathrm{~kg}\) an astronaut throws a rock of mass 2.00 kg straight upward with an initial speed \(40.0 \mathrm{~m} / \mathrm{s}\). (This moon is too small to have an atmosphere.) What maximum height above the surface of the moon will the rock reach?
See all solutions

Recommended explanations on Physics Textbooks

Waves Physics

Read Explanation

Conservation of Energy and Momentum

Read Explanation

Work Energy and Power

Read Explanation

Space Physics

Read Explanation

Circular Motion and Gravitation

Read Explanation

Engineering Physics

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