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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.

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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.

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Most popular questions from this chapter

Object \(\mathrm{A}\) is moving due east, while object \(\mathrm{B}\) is moving due north. They collide and stick together in a completely inelastic collision. Momentum is conserved. (a) Is it possible that the two-object system has a final total momentum of zero after the collision? (b) Roughly, what is the direction of the final total momentum of the two-object system after the collision?

The drawing shows a human figure approximately in a sitting position. For purposes of this problem, there are three parts to the figure, and the center of mass of each one is shown in the drawing. These parts are: (1) the torso, neck, and head (total mass \(=41 \mathrm{~kg}\) ) with a center of mass located on the \(y\) axis at a point \(0.39 \mathrm{~m}\) above the origin, (2) the upper legs (mass \(=17 \mathrm{~kg}\) ) with a center of mass located on the \(x\) axis at a point \(0.17 \mathrm{~m}\) to the right of the origin, and (3) the lower legs and feet (total mass \(=9.9 \mathrm{~kg}\) ) with a center of mass located \(0.43 \mathrm{~m}\) to the right of and \(0.26 \mathrm{~m}\) below the origin. Find the \(x\) and \(y\) coordinates of the center of mass of the human figure. Note that the mass of the arms and hands (approximately \(12 \%\) of the wholebody mass) has been ignored to simplify the drawing.

A two-stage rocket moves in space at a constant velocity of \(4900 \mathrm{~m} / \mathrm{s} .\) The two stages are then separated by a small explosive charge placed between them. Immediately after the explosion the velocity of the 1200 -kg upper stage is \(5700 \mathrm{~m} / \mathrm{s}\) in the same direction as before the explosion. What is the velocity (magnitude and direction) of the 2400 -kg lower stage after the explosion?

A 50.0 -kg skater is traveling due east at a speed of \(3.00 \mathrm{~m} / \mathrm{s}\). A \(70.0-\mathrm{kg}\) skater is moving due south at a speed of \(7.00 \mathrm{~m} / \mathrm{s}\). They collide and hold on to each other after the collision, managing to move off at an angle \(\theta\) south of east, with a speed of \(v_{\mathrm{f}}\). Find (a) the angle \(\theta\) and (b) the speed \(v_{\mathrm{f}}\), assuming that friction can be ignored.

A ball is attached to one end of a wire, the other end being fastened to the ceiling. The wire is held horizontal, and the ball is released from rest (see the drawing). It swings downward and strikes a block initially at rest on a horizontal frictionless surface. Air resistance is negligible, and the collision is elastic. (a) During the downward motion of the ball, are any of the following conserved: its momentum, its kinetic energy, its total mechanical energy? (b) During the collision with the block, are any of the following conserved: the horizontal component of the total momentum of the ball/block system, the total kinetic energy of the system? Provide reasons for your choices.
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