/*! 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 72 Find the speed of a particle who... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 33: Problem 72

Find the speed of a particle whose relativistic kinetic energy is \(50 \%\) greater than the Newtonian value calculated for the same speed.

Short Answer

Expert verified
The speed can be found using the above steps, and entails the solution of a quadratic equation, of which only the solution which is less than or equal to the speed of light is valid.

Step by step solution

01

Set up the equation for Newtonian Kinetic Energy

Damage the Newtonian kinetic energy, which is given by the formula \( K_{N} = \frac{1}{2} mv^2 \). It is assumed that the mass \( m \) is known, and the speed \( v \) is what we're trying to find.
02

Set up the equation for Relativistic Kinetic Energy

The Lorentz factor \( \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \), where \( c \) is the speed of light, is used to calculate the relativistic kinetic energy. The relativistic kinetic energy formula is \( K_{rel} = mc^2(\gamma - 1) \). Remember that our aim is to find the speed \( v \).
03

Set up the relation between Relativistic and Newtonian Kinetic Energies

We are given that the relativistic kinetic energy is 50% greater than the Newtonian kinetic energy, which leads to \( K_{rel} = 1.5K_{N} \). Substitute the formulas of \( K_{rel} \) and \( K_{N} \) from the first and second stages into this equation.
04

Solve for speed

After substituting into the equation in the third step, we get a quadratic equation in terms of \( v \). Solve this quadratic equation using the quadratic formula. The valid solution is the one where \( v \leq c \), because the speed cannot be greater than 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!

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

By what factor does an object's momentum change if you double its speed when its original speed is (a) \(25 \mathrm{m} / \mathrm{s}\) and (b) \(100 \mathrm{Mm} / \mathrm{s} ?\)

You've been named captain of NASA's first interstellar mission since the Voyager robotic spacecraft. You board your spaceship. accelerate quickly to \(0.8 c,\) and cruise at constant speed toward Proxima Centauri, the closest star to our Sun. Proxima Centauri is 4 light-years distant as measured in the two stars' common rest frame. On the way, you conduct various medical experiments to determine the effects of a long space voyage on the human body. Taking your pulse, you find a. it's significantly slower than when you're on Earth. b. it's the same as when you're on Earth. c. it's significantly faster than when you're on Earth.

A hospital's linear accelerator produces electron beams for cancer treatment. The accelerator is \(1.6 \mathrm{m}\) long and the electrons reach a speed of \(0.98 c .\) How long is the accelerator in the electrons" reference frame?

Twins \(A\) and \(B\) live on Earth. On their 20 th birthday, twin \(B\) climbs into a spaceship and makes a round-trip journey at \(0.95 c\) to a star 30 light years distant, as measured in the Earth-star reference frame. What are their ages when twin B returns to Earth?

A spaceship passes by you at half the speed of light, and you determine that it's \(35 \mathrm{m}\) long. Find its length as measured in its rest frame.
See all solutions

Recommended explanations on Physics Textbooks

Classical Mechanics

Read Explanation

Atoms and Radioactivity

Read Explanation

Measurements

Read Explanation

Conservation of Energy and Momentum

Read Explanation

Medical Physics

Read Explanation

Fundamentals of Physics

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.