Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 26: Problem 57

Starting with the energy-momentum relation \(E^{2}=E_{0}^{2}+\) \((p c)^{2}\) and the definition of total energy, show that $(p c)^{2}=K^{2}+2 K E_{0}[\mathrm{Eq} .(26-11)]$.

Short Answer

Expert verified
Question: Derive the relation \((pc)^2 = K^2 + 2KE_0\) using the energy-momentum relation \(E^2 = E_0^2 + (pc)^2\) and the definition of total energy. Answer: \((pc)^2 = K^2 + 2KE_0\)

Step by step solution

01

Express total energy in terms of kinetic energy and rest energy

Total energy (E) is the sum of kinetic energy (K) and rest energy (\(E_0\)) for a particle: $$ E = K + E_0 $$
02

Substitute the expression for total energy into the energy-momentum relation

We have the energy-momentum relation: $$ E^2 = E_0^2 + (pc)^2 $$ Substitute \(E = K + E_0\) into the relation: $$ (K + E_0)^2 = E_0^2 +(pc)^2 $$
03

Expand and simplify

Expand the left side of the equation: $$ (K^2 + 2KE_0 + E_0^2) = E_0^2 + (pc)^2 $$ Subtract \(E_0^2\) from both sides: $$ K^2 + 2KE_0 = (pc)^2 $$
04

Final result

We have shown that using the energy-momentum relation and the definition of total energy, we can derive the equation: $$ (pc)^2 = K^2 + 2KE_0 $$ This is the desired result.

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

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

The Tevatron is a particle accelerator at Fermilab that accelerates protons and antiprotons to high energies in an underground ring. Scientists observe the results of collisions between the particles. The protons are accelerated until they have speeds only \(100 \mathrm{m} / \mathrm{s}\) slower than the speed of light. The circumference of the ring is \(6.3 \mathrm{km} .\) What is the circumference according to an observer moving with the protons? [Hint: Let \(v=c-u\) where $v \text { is the proton speed and } u=100 \mathrm{m} / \mathrm{s} .$]
Two spaceships are observed from Earth to be approaching each other along a straight line. Ship A moves at \(0.40 c\) relative to the Earth observer, and ship \(\mathrm{B}\) moves at \(0.50 c\) relative to the same observer. What speed does the captain of ship A report for the speed of ship B?
A lambda hyperon \(\Lambda^{0}\) (mass \(=1115 \mathrm{MeV} / c^{2}\) ) at rest decays into a neutron \(\mathrm{n}\) (mass \(=940 \mathrm{MeV} / \mathrm{c}^{2}\) ) and a pion (mass \(=135 \mathrm{MeV} / c^{2}\)): $$\Lambda^{0} \rightarrow \mathrm{n}+\pi^{0}$$ What is the total kinetic energy of the neutron and pion?
When an electron travels at \(0.60 c,\) what is its total energy in mega- electron-volts?
An unstable particle called the pion has a mean lifetime of 25 ns in its own rest frame. A beam of pions travels through the laboratory at a speed of $0.60 c .$ (a) What is the mean lifetime of the pions as measured in the laboratory frame? (b) How far does a pion travel (as measured by laboratory observers) during this time?
See all solutions

Recommended explanations on Physics Textbooks

Further Mechanics and Thermal Physics

Read Explanation

Waves Physics

Read Explanation

Particle Model of Matter

Read Explanation

Linear Momentum

Read Explanation

Solid State Physics

Read Explanation

Dynamics

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.

Sign-up for free