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Chapter 4: Problem 30

Two comets are leaving the vicinity of the Sun, one traveling in an elliptical orbit and the other in a hyperbolic orbit. What can you say about the future of these two comets? Would you expect either of them to return eventually?

Short Answer

Expert verified
The elliptical orbit comet will return; the hyperbolic orbit comet will not.

Step by step solution

01

Identify the Types of Orbits

Recognize that the two comets are in different types of orbits: one in an elliptical orbit and the other in a hyperbolic orbit.
02

Characteristics of an Elliptical Orbit

Understand that an elliptical orbit is a closed orbit. This means the comet will travel around the Sun and return periodically.
03

Characteristics of a Hyperbolic Orbit

Understand that a hyperbolic orbit is an open orbit. The comet in a hyperbolic orbit will escape the gravitational pull of the Sun and will not return.
04

Conclusion on Return to the Sun

Based on the characteristics of the orbits, the comet in the elliptical orbit will return to the vicinity of the Sun, while the comet in the hyperbolic orbit will not return.

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Key Concepts

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

Elliptical Orbits
An elliptical orbit is a closed path around an object, like the Sun. When a comet follows an elliptical orbit, it means the comet will consistently travel around the Sun, coming closer and then moving farther away, before repeating this cycle. Imagine an oval shape - that's the basic form of an elliptical orbit.
Some key characteristics include:
  • Regular Return: Because the orbit is closed, the comet will always come back to the point near the Sun where it started.
  • Varying Speed: The comet travels faster when it's closer to the Sun and slower when it's farther away.
Therefore, a comet in an elliptical orbit will eventually return to the Sun鈥檚 vicinity.
Hyperbolic Orbits
A hyperbolic orbit, in contrast, is an open path. This means it's like a one-way ticket for the comet out of the solar system. The shape is more like a stretched-out curve, diverging forever.
Here are a few key points:
  • Non-return Path: Unlike elliptical orbits, once a comet enters a hyperbolic orbit, it will escape the gravitational pull of the Sun and won't return.
  • Originating Outside the Solar System: Often, comets in hyperbolic orbits are thought to come from outside our solar system, making a brief appearance near the Sun before leaving forever.
So, when a comet moves in a hyperbolic orbit, it is essentially saying goodbye to the Sun for good.
Gravitational Pull
Gravity plays a crucial role in shaping both elliptical and hyperbolic orbits. The gravitational pull of the Sun determines how these orbits form and behave.
Here are the essentials to understand:
  • Binding Force: For elliptical orbits, the Sun鈥檚 gravity acts as a continuous binding force, pulling the comet back each time it tries to move away.
  • Escape Velocity: For a comet to enter a hyperbolic orbit, it must reach a speed high enough to overcome the Sun鈥檚 gravitational pull completely. This speed is known as the escape velocity.
In short, the gravitational pull is what keeps the comet in an elliptical orbit or allows it to break free into a hyperbolic orbit.

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Most popular questions from this chapter

\(\mathbf{T} / \mathbf{F}:\) When an object falls to Earth, Earth also falls up to the object.

Earth's mean radius and mass are \(6,370 \mathrm{km}\) and \(5.97 \times 10^{24} \mathrm{kg}\) respectively. Show that the acceleration of gravity at the surface of Earth is \(9.81 \mathrm{m} / \mathrm{s}^{2}\)

Earth speeds along at \(29.8 \mathrm{km} / \mathrm{s}\) in its orbit. Neptune's nearly circular orbit has a radius of \(4.5 \times 10^{9} \mathrm{km},\) and the planet takes 164.8 years to make one trip around the Sun. Calculate the speed at which Neptune plods along in its orbit.

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Rank the following objects, in order of their circular velocities, from smallest to largest. a. a \(5-\mathrm{kg}\) object orbiting Earth halfway to the Moon b. a \(10-\mathrm{kg}\) object orbiting Earth just above Earth's surface c. a 15 -kg object orbiting Earth at the same distance as the Moon d. a 20 -kg object orbiting Earth one-quarter of the way to the Moon
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