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Chapter 4: Problem 32

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 moves along in its orbit.

Short Answer

Expert verified
5.44 km/s

Step by step solution

01

Understand the Problem

We are given the orbital speed of Earth and need to calculate the orbital speed of Neptune. We have the radius of Neptune's orbit and the time it takes to complete one orbit.
02

Write Down Known Values

Neptune's orbital radius, \( R = 4.5 \times 10^9 \text{ km} \) Orbital period of Neptune, \( T = 164.8 \text{ years} \)
03

Convert Period to Seconds

There are 365.25 days in a year, 24 hours in a day, 60 minutes in an hour, and 60 seconds in a minute. Convert the period from years to seconds using these conversions. \[T = 164.8 \text{ years} \times 365.25 \text{ days/year} \times 24 \text{ hours/day} \times 60 \text{ minutes/hour} \times 60 \text{ seconds/minute}\]\[T \text{ in seconds} = 164.8 \times 365.25 \times 24 \times 60 \times 60\]\[ T ≈ 5.1934 \times 10^9 \text{ seconds} \]
04

Calculate Neptune's Orbital Speed

The formula for the speed of an object in a circular orbit is given by the circumference of the orbit divided by the orbital period. \[\text{Speed} = \frac{2 \times \text{π} \times R}{T}\]Substitute the known values: \[\text{Speed} = \frac{2 \times π \times 4.5 \times 10^9 \text{ km}}{5.1934 \times 10^9 \text{ seconds}}\]\[\text{Speed} ≈ \frac{28.274 \times 10^9 \text{ km}}{5.1934 \times 10^9 \text{ seconds}} \]\[\text{Speed} ≈ 5.44 \text{ km/s} \]

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Key Concepts

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

Kepler's Laws
Johannes Kepler's laws of planetary motion are fundamental to understanding how planets move in our solar system. There are three key laws to remember:
  • **First Law (Law of Ellipses):** Planets orbit the sun in elliptical paths with the Sun at one of the two foci.
  • **Second Law (Law of Equal Areas):** A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time. This means that planets move faster when they are closer to the Sun and slower when they are further away.
  • **Third Law (Law of Harmonies):** The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit. Mathematically, it can be expressed as \(\frac{T^2}{R^3} = \text{constant}\), where \(T\) is the orbital period and \(R\) is the semi-major axis's length.
These laws help us calculate various properties of planetary motion, including orbital speed.
Orbital Mechanics
Orbital mechanics is the study of the motions of natural and artificial celestial bodies. It is governed by Kepler's laws and Newton’s laws of motion and gravitation. Here's a breakdown of the main principles:
  • **Gravitational Force:** This force is what keeps planets in orbit around the Sun. Newton's law of gravitation states that every mass exerts an attractive force on every other mass. This force decreases with the square of the distance between the centers of the two masses.
  • **Circular Orbits:** For our calculations, we often simplify the problem by assuming circular orbits. The orbital speed can be calculated by dividing the orbit's circumference by the planet’s orbital period.
  • **Equations of Motion:** The equations for determining orbital properties often involve trigonometric functions and calculus. However, for circular orbits, simpler formulas can be used, such as the speed formula derived in the exercise: \( \text{Speed} = \frac{2 \times \text{π} \times R}{T} \), where \( R \) is the radius of the orbit and \( T \) is the orbital period.
Understanding these principles allows us to grasp the dynamics of planetary motion efficiently.
Planetary Motion
Planetary motion refers to the movement of planets around the Sun and can be understood through both observational data and theoretical principles.
  • **Elliptical Orbits:** While we often assume circular orbits for simplicity, it's crucial to remember that actual orbits are elliptical, as specified by Kepler’s first law.
  • **Orbital Period:** The time it takes for a planet to make one complete orbit around the Sun. This period is influenced by the planet's distance from the Sun and provides crucial data for calculating orbital speed.
  • **Orbital Speed:** This is the rate at which a planet travels along its orbit. For circular orbits, the speed can be calculated using the formula provided in the exercise. This speed ensures that the planet remains in a stable orbit due to the balance between gravitational pull and its inertial motion.
  • **Planetary Influences:** Factors like the planet's mass, the gravitational pull of other celestial bodies, and axial tilt can influence planetary motion. However, for fundamental calculations, we assume ideal conditions to simplify the process.
Using these concepts, we can calculate Neptune's orbital speed or that of any other planet, providing a clearer understanding of our solar system's dynamics.

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Most popular questions from this chapter

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

The connection between gravity and orbits enables astronomers to measure the ________ of stars and planets. a. distances b. sizes c. masses d. compositions

a. Watch the first 12 minutes of the episode "Gravity" in the Universe series (http://www.history.com/shows/theuniverse/videos/the-universe-gravity) to see several illustrated examples of gravity. Why does the tennis ball appear to float when dropped at the top of the roller coaster? How was Newton able to imagine a satellite orbiting Earth centuries before it was possible? Why is it the speed of the cannonball that determines whether it goes into orbit? What was the technical difficulty in launching a satellite? b. In the same video as in part (a), watch the trip on the zero-G plane, at \(23-29\) minutes. How does the plane simulate zero-G? Why does it last for only \(20-30\) seconds? How is this similar to the roller coaster in part (a)?

Compared to your mass on Earth, on the Moon your mass would be a. lower because the Moon is smaller than Earth. b. lower because the Moon has less mass than Earth. c. higher because of the combination of the Moon's mass and size. d. the same, mass doesn't change.

Suppose you go skydiving. a. Just as you fall out of the airplane, what is your gravitational acceleration? b. Would this acceleration be bigger, smaller, or the same if you were strapped to a flight instructor, and so had twice the mass? c. Just as you fall out of the airplane, what is the gravitational force on you? (Assume your mass is \(70 \mathrm{kg}\).) d. Would the gravitational force be bigger, smaller, or the same if you were strapped to a flight instructor, and so had twice the mass?
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