/*! 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 21 ssm During July 1994 the comet S... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 21

ssm During July 1994 the comet Shoemaker-Levy 9 smashed into Jupiter in a spectacular fashion. The comet actually consisted of 21 distinct pieces, the largest of which had a mass of approximately \(4.0 \times 10^{12} \mathrm{~kg}\) and a speed of \(6.0 \times 10^{4} \mathrm{~m} / \mathrm{s}\). Jupiter, the largest planet in the solar system, has a mass of \(1.9 \times 10^{27} \mathrm{~kg}\) and an orbital speed of \(1.3 \times 10^{4} \mathrm{~m} / \mathrm{s}\). If this piece of the comet had hit Jupiter head-on, what would have been the change (magnitude only) in Jupiter's orbital speed (not its final speed)?

Short Answer

Expert verified
The change in Jupiter's orbital speed is approximately \(1.3 \times 10^{-15} \text{ m/s}\).

Step by step solution

01

Understand the Conservation of Momentum

To find the change in Jupiter's orbital speed after the comet hits, we have to apply the principle of conservation of momentum. The total momentum before the collision must equal the total momentum after the collision because the system is closed and there are no external forces.
02

Write Down Known Values

List the known values: the mass of the comet piece is \( m_c = 4.0 \times 10^{12} \text{ kg} \) and its speed is \( v_c = 6.0 \times 10^4 \text{ m/s} \). The mass of Jupiter is \( m_j = 1.9 \times 10^{27} \text{ kg} \) and its initial orbital speed is \( v_j = 1.3 \times 10^4 \text{ m/s} \).
03

Set Up the Momentum Equation

Use the conservation of momentum equation for a head-on collision:\[m_c v_c + m_j v_j = (m_c + m_j) v_{fj}\]where \( v_{fj} \) is the final velocity of Jupiter after impact.
04

Solve for Final Velocity

Rearranging the equation to solve for \( v_{fj} \):\[v_{fj} = \frac{m_c v_c + m_j v_j}{m_c + m_j}\]Substitute the known values:\[v_{fj} = \frac{(4.0 \times 10^{12} \text{ kg})(6.0 \times 10^4 \text{ m/s}) + (1.9 \times 10^{27} \text{ kg})(1.3 \times 10^4 \text{ m/s})}{4.0 \times 10^{12} \text{ kg} + 1.9 \times 10^{27} \text{ kg}}\]
05

Calculate and Determine the Change in Speed

Calculating the expression will give the final speed \( v_{fj} \) and the change in Jupiter's speed is:\[\Delta v = v_{fj} - v_j\]Since we are looking for the magnitude of the change, calculate and determine the difference.
06

Compute Result

Plug into a calculator:The total momentum before impact is approximately \( 4.72 \times 10^{31} \) and the mass after impact is \( 1.9 \times 10^{27} \). Solve for \( v_{fj} \) to find the value, and compute \( \Delta v \approx 1.3 \times 10^{-15} \text{ m/s} \) for just the portion caused by the comet's impact.

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.

Head-On Collision
A head-on collision refers to an event where two objects collide directly with one another along the same line of motion. In physics, this is a type of
  • collision that often simplifies calculations,
  • especially when using the conservation of momentum.

The speed and direction of the involved objects before and after the collision can be analyzed using this principle.
In the case of Shoemaker-Levy 9 and Jupiter, the comet's path was directly aligned with Jupiter, making their encounter a perfect example of a head-on collision. Understanding head-on collisions helps in calculating the resultant velocities and understanding the impact of forces when compared to glancing or side collisions.
Orbital Speed
Orbital speed is the constant speed at which a celestial body, like a planet or moon, travels along its orbit around another body. This can be seen in planets
  • orbiting the sun, moons orbiting planets, and
  • man-made satellites orbiting the Earth.

The orbital speed ensures that the object remains in its path due to the balance between gravitational pull and its velocity.
Jupiter's orbital speed is initially given as \(1.3 \times 10^4 \text{ m/s}\), showcasing how massive and fast it is relative to the rest of the solar system. When an external force, like a comet's impact, is introduced, these speeds can minimally adjust while still conserving the overall momentum of the system.
Mass of Celestial Bodies
Celestial bodies like planets, stars, and moons have enormous masses that play a critical role in gravitational attractions and celestial mechanics.
  • Jupiter is the largest planet in our solar system with a mass of \(1.9 \times 10^{27} \text{ kg}\), which
  • means it has a substantial gravitational influence over its moons and occasionally passing celestial objects like comets.

This enormous mass makes the change in velocity (\(\Delta v\)) from external forces, like a comet collision, negligible when considered in the grand scale of the solar system.
However, even a relatively small impact can provide us data to understand these interactions better and calculate minute changes in momentum and velocity as demonstrated by the Shoemaker-Levy 9 event.
Change in Velocity
The change in velocity, often symbolized as \(\Delta v\), is a crucial measure in physics. It helps understand how an interaction affects the motion of objects in space.
  • In the context of the Shoemaker-Levy 9 impact, we calculate the change in Jupiter's velocity due to
  • the kinetic energy transferred from the comet to the planet.

Despite Jupiter's massive scale, the comet still imparts a change, albeit tiny, calculated to be \(1.3 \times 10^{-15} \text{ m/s}\).
This showcases that even in astronomical terms, every collision has quantifiable effects. Understanding and calculating this change helps us grasp how celestial events influence a planet's movement across its vast orbital path.

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

A golf ball strikes a hard, smooth floor at an angle of \(30.0^{\circ}\) and, as the drawing shows, rebounds at the same angle. The mass of the ball is \(0.047 \mathrm{~kg},\) and its speed is \(45 \mathrm{~m} / \mathrm{s}\) just before and after striking the floor. What is the magnitude of the impulse applied to the golf ball by the floor?

A person stands in a stationary canoe and throws a \(5.00-\mathrm{kg}\) stone with a velocity of 8.00 \(\mathrm{m} / \mathrm{s}\) at an angle of \(30.0^{\circ}\) above the horizontal. The person and canoe have a combined mass of \(105 \mathrm{~kg}\). Ignoring air resistance and effects of the water, find the horizontal recoil velocity (magnitude and direction) of the canoe.

At in preparation for this problem. Two friends, Al and Jo, have a combined mass of \(168 \mathrm{~kg}\). At an ice skating rink they stand close together on skates, at rest and facing each other, with a compressed spring between them. The spring is kept from pushing them apart because they are holding each other. When they release their arms, Al moves off in one direction at a speed of \(0.90 \mathrm{~m} / \mathrm{s}\), while Jo moves off in the opposite direction at a speed of \(1.2 \mathrm{~m} / \mathrm{s}\). Assuming that friction is negligible, find Al's mass.

When jumping straight down, you can be seriously injured if you land stiff- legged One way to avoid injury is to bend your knees upon landing to reduce the force of the impact. A 75 -kg man just before contact with the ground has a speed of \(6.4 \mathrm{~m} / \mathrm{s}\). (a) In a stiff-legged landing he comes to a halt in \(2.0 \mathrm{~ms}\). Find the average net force that acts on him during this time. (b) When he bends his knees, he comes to a halt in \(0.10 \mathrm{~s}\). Find the average net force now. (c) During the landing, the force of the ground on the man points upward, while the force due to gravity points downward. The average net force acting on the man includes both of these forces. Taking into account the directions of the forces, find the force of the ground on the man in parts (a) and (b).

A \(0.500-\mathrm{kg}\) ball is dropped from rest at a point \(1.20 \mathrm{~m}\) above the floor. The ball rebounds straight upward to a height of \(0.700 \mathrm{~m}\). What are the magnitude and direction of the impulse of the net force applied to the ball during the collision with the floor?
See all solutions

Recommended explanations on Physics Textbooks

Conservation of Energy and Momentum

Read Explanation

Translational Dynamics

Read Explanation

Thermodynamics

Read Explanation

Further Mechanics and Thermal Physics

Read Explanation

Astrophysics

Read Explanation

Atoms and Radioactivity

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.