/*! 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 14 How does modem relativity modify... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 14

How does modem relativity modify the law of conservation of momentum?

Short Answer

Expert verified
In the context of special relativity, the law of conservation of momentum is modified using the relativistic momentum formula, given by \(\vec{p} = \dfrac{m\vec{v}}{\sqrt{1 - \frac{v^2}{c^2}}}\), where \(m\) is the rest mass of the particle, \(v\) is its velocity, and \(c\) is the speed of light in a vacuum. Consequently, the relativistic law of conservation of momentum is expressed as \(危 \dfrac{m\vec{v}}{\sqrt{1 - \frac{v^2}{c^2}}} = 危 \dfrac{m'\vec{v'}}{\sqrt{1 - \frac{{v'}^2}{c^2}}}\), demonstrating that the conservation of momentum still holds true even as particles approach the speed of light, with their mass effectively increasing.

Step by step solution

01

Classical Law of Conservation of Momentum

In classical physics, the law of conservation of momentum states that the total momentum of a closed system of particles is conserved over time, provided that no external forces are acting on the system. Mathematically, it can be represented as: \(危\vec{p} = 危\vec{p'}\) Here, 危 represents the summation over all particles in the system, \(\vec{p}\) is the initial momentum, and \(\vec{p'}\) is the final momentum.
02

Relativistic Momentum

According to the special theory of relativity, as a particle's speed approaches the speed of light, its mass effectively increases and thus, momentum is no longer given by the classical formula: \(\vec{p} = m\vec{v}\) Instead, it is given by the relativistic momentum formula: \(\vec{p} = \dfrac{m\vec{v}}{\sqrt{1 - \frac{v^2}{c^2}}}\) where \(m\) is the rest mass of the particle, \(v\) is the velocity, and \(c\) is the speed of light in a vacuum.
03

Relativistic Law of Conservation of Momentum

In special relativity, the conservation of momentum is similarly based on the principle that the total momentum of an isolated system remains constant. However, it must now be expressed using the relativistic momentum formula: \(危 \dfrac{m\vec{v}}{\sqrt{1 - \frac{v^2}{c^2}}} = 危 \dfrac{m'\vec{v'}}{\sqrt{1 - \frac{{v'}^2}{c^2}}}\) Here, 危 represents the summation over all particles in the system, \(m\vec{v}\) and \(m'\vec{v'}\) are the initial and final momenta of the particles, and \(c\) is the speed of light. This equation demonstrates that even in the context of special relativity, the law of conservation of momentum still holds true, though it must be expressed using the modified relativistic momentum formula to account for the increasing mass and corresponding momentum of particles as they approach the speed of light.

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.

Conservation of Momentum
The concept of momentum conservation is crucial in understanding physical interactions. In classical physics, momentum is simply the product of mass and velocity. The law of conservation of momentum tells us that in a closed system with no external forces, the total momentum before any interaction is equal to the total momentum after.
This principle ensures that motion changes in a predictable way, adhering to the equation: \[危\vec{p} = 危\vec{p'}\]where \(\vec{p}\) is the momentum vector.
Momentum conservation helps us solve a myriad of physics problems, like collisions, by analyzing the system's initial and final states.
Special Theory of Relativity
Developed by Einstein, the Special Theory of Relativity revolutionized our understanding of space and time. A core idea is that the laws of physics are the same for all observers, regardless of their relative velocity. This theory led to new insights into how objects behave at high speeds, especially those close to the speed of light.
One of the significant impacts of relativity is the concept of relativistic momentum. As an object's speed approaches light speed, its mass appears to increase from the perspective of an external observer. Thus, the classical formula for momentum \(m\vec{v}\) becomes insufficient. The relativistic momentum formula is:
  • \(\vec{p} = \frac{m\vec{v}}{\sqrt{1 - \frac{v^2}{c^2}}}\)
  • Where \(m\) is rest mass and \(c\) is the speed of light.
This adjustment ensures that the classical laws modify as needed at very high velocities, keeping the core physics principles consistent.
Speed of Light
In physics, the speed of light \(c\) is a fundamental constant, approximately valued at 299,792,458 meters per second in a vacuum. It acts as a cosmic speed limit, laying the groundwork for the upper bounds of speed for particles according to Einstein's theories.
The speed of light is not just a speed, but a limit beyond which no object with mass can travel. This boundary has profound implications, influencing how we understand time, space, and energy exchange.
For instance, as an object accelerates closer to \(c\), its relativistic effects become more prominent 鈥� its mass appears to increase asymptotically. This indicates that infinite energy would be required to accelerate a massive object to light speed, thus enforcing \(c\) as the ultimate speed barrier in the universe.
Therefore, understanding \(c\) allows us to reconcile the behaviors observed in high-speed particle experiments with theoretical predictions.

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) Calculate the speed of a 1.00 - \(\mu\) g particle of dust that has the same momentum as a proton moving at \(0.999 c\) (b) What does the small speed tell us about the mass of a proton compared to even a tiny amount of macroscopic matter?

What is the rest energy of an electron, given its mass is \(9.11 \times 10^{-31} \mathrm{kg} ?\) Give your answer in joules and \(\mathrm{MeV}\)

All galaxies farther away than about \(50 \times 10^{6}\) ly exhibit a red shift in their emitted light that is proportional to distance, with those farther and farther away having progressively greater red shifts. What does this imply, assuming that the only source of red shift is relative motion?

(a) At what relative velocity is \(\gamma=1.50 ?\) (b) At what relative velocity is \(\gamma=100 ?\)

Show that the relativistic form of Newton's second law is (a) \(F=m \frac{d u}{d t} \frac{1}{\left(1-u^{2} / c^{2}\right)^{3 / 2}} ;\) (b) Find the force needed to accelerate a mass of \(1 \mathrm{kg}\) by \(1 \mathrm{m} / \mathrm{s}^{2}\) when it is traveling at a velocity of \(c / 2\).
See all solutions

Recommended explanations on Physics Textbooks

Electromagnetism

Read Explanation

Physical Quantities And Units

Read Explanation

Force

Read Explanation

Electric Charge Field and Potential

Read Explanation

Modern Physics

Read Explanation

Geometrical and Physical Optics

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.