/*! 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 13 Which of the following is true c... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 13

Which of the following is true concerning the spacecraft and the moon when they are in the same orbit? (Assume that neither is using a propulsion system to maintain its orbit.) A. They both must be at the same speed. B. They both must have the same mass. C. They both must have the same mass and speed. D. They must have different masses.

Short Answer

Expert verified
A. They both must be at the same speed.

Step by step solution

01

Understanding Orbital Mechanics

In an orbit, the speed of an object is determined by the balance between gravitational force and centripetal force. The gravitational force depends on the mass of the central body (e.g., a planet or star) and the distance of the object from this body.
02

Analyzing the Given Scenario

The spacecraft and the moon are both in the same orbit, circling a central body like the Earth. This means they are at the same distance from the central body.
03

Applying the Orbital Speed Concept

For any two objects in the same orbit (same distance from the central body), the orbital speed is given by the formula: \( v = \sqrt{ \frac{GM}{r} } \), where \( G \) is the gravitational constant, \( M \) is the mass of the central body, and \( r \) is the radius of the orbit. Since \( r \) (orbit radius) and \( M \) (mass of central body) are constants for both objects, both the spacecraft and moon must move at the same speed to maintain the orbit.
04

Evaluating the Options

A. True. Both must be at the same speed to maintain the same orbit. B. False. The mass of the objects doesn't affect their speed in orbit. C. False. While they do have the same speed, their masses can be different. D. False. They can have different masses, but this doesn't affect them maintaining the same orbit.
05

Conclusion

Based on the analysis, the correct answer is A. They both must be at the same speed.

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.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Gravitational Force
Gravitational force is the attractive force between two masses. In the context of orbital mechanics, it's the force that keeps planets, moons, and spacecraft in their orbits. According to Newton's law of universal gravitation, the force (\f) between two objects is given by: \[ F = G \frac{m1 \times m2}{r^2} \] where:
  • G is the gravitational constant,
  • m1 and m2 are the masses of the two objects,
  • r is the distance between the centers of the two objects.
This equation shows that the force increases with larger masses and decreases with greater distance. In orbital mechanics, one of the masses is usually a large celestial body like Earth, and the other is a smaller object like a moon or spacecraft.The balance between this gravitational force and the object's inertia keeps it in orbit. Because the gravitational force is central, it acts as a centripetal force, pulling the orbiting object towards the center of the large celestial body.
Centripetal Force
Centripetal force is the force that keeps an object moving in a circular path. It's directed towards the center of the circle and its magnitude is given by: \[ F_c = \frac{mv^2}{r} \] where:
  • m is the mass of the object,
  • v is its speed,
  • r is the radius of the circular path.
In the context of an orbit, the centripetal force needed to keep the object moving in a circle is provided by the gravitational force. This means: \[ G \frac{m1 \times m2}{r^2} = \frac{m2 \times v^2}{r} \] Solving for speed (v) gives us the orbital speed formula, which shows that speed depends only on the mass of the central body and the radius of the orbit, not on the mass of the orbiting object. This explains why a spacecraft and a moon in the same orbit must have the same speed despite their different masses.
Orbital Speed
Orbital speed is the speed an object must have to stay in orbit around a central body without propulsion. It balances the gravitational pull of the central body and the inertia of the object moving along a curved path. The orbital speed (v) for an object in a circular orbit is given by: \[ v = \frac{GM}{r} \] where:
  • G is the gravitational constant,
  • M is the mass of the central body,
  • r is the radius of the orbit.
From the formula, you can see that orbital speed depends on the mass of the central body and the distance from it, but not on the mass of the orbiting object. For our specific case, since both the spacecraft and the moon are at the same distance from the Earth, they both require the same speed to stay in that shared orbit. This means regardless of their mass, as long as they are at the same orbital radius, they maintain the same speed.

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

The space ship experiences centripetal acceleration while orbiting the planet. According to Newton's laws of motion, if the spaceship encounters no resisting force in the course of its circular orbit, what will be its future path? A. It will orbit in a circle forever. B. It will gradually spiral inward. C. It will gradually spiral outward. D. It will break from the orbit to travel in a straight line.

The passage comes from the imagination of a student. A real pendulum on the clock in orbit would: A. swing more slowly than it would if it were on the planet below. B. swing more swiftly than it would if it were on the planet below. C. swing at the same rate as it would if it were on the planet below. D. not swing on the orbiting spacecraft.

If \(F\) is the gravitational force created on the moon by the earth, which of the following expressions is equal to the gravitational force created on the earth by the moon? A. \(F\) B. \(\frac{\left(5.97 \times 10^{24}\right) \times F}{\left(7.5 \times 10^{22}\right)}\) C. \(\frac{\left(7.5 \times 10^{22}\right) \times F}{\left(5.97 \times 10^{24}\right)}\) D. \(\frac{(1,738)^2 \times F}{\left(5.97 \times 10^{24}\right)\left(7.5 \times 10^{22}\right)}\)

All of the following will affect the time of flight for a projectile experiencing no air resistance EXCEPT. I. the mass of the projectile II. the initial horizontal velocity of the projectile III. the initial vertical velocity of the projectile A. I only B. III only C. I and II oniy D. I and III only

A lunar day is defined as the time that elapses from sunrise to the following sunrise on the moon at a given location. How long is one lunar day? A. 12 earth hours B. 24 earth hours C. \(27.3\) earth days D. one earth year
See all solutions

Recommended explanations on Physics Textbooks

Modern Physics

Read Explanation

Dynamics

Read Explanation

Geometrical and Physical Optics

Read Explanation

Turning Points in Physics

Read Explanation

Magnetism

Read Explanation

Classical Mechanics

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.