COLLEGE PHYSICS,VOLUME 1
COLLEGE PHYSICS,VOLUME 1
2nd Edition
ISBN: 9781319115104
Author: Freedman
Publisher: MAC HIGHER
bartleby

Videos

Question
Book Icon
Chapter 25, Problem 77QAP
To determine

(a)

Does the muon reach Earth's surface within one half-life if Muons have a proper half-life of 1.56×106s, suppose a muon is formed at an altitude of 3000 m and travels at a speed of 0.950c straight toward Earth? Complete the problem from both perspectives: the muon's point of view and Earth's reference flame.

Expert Solution
Check Mark

Answer to Problem 77QAP

The muon does not make it to Earth

Explanation of Solution

Calculation:

Muons have a proper lifetime of Δtproper=1.56×106s. A muon is formed a distance of 3000 m above the surface of Earth and travels straight downward with a speed V = 0.950c. To determine whether or not the muon reaches Earth's surface before it decays, we have to calculate how far the muon travels in 1.56×106srelative to both the reference frame of the muon and the reference frame of Earth. The distance the muon travels will be contracted in the muon's reference frame, but the observed time will be longer in Earth's reference frame. In either case, the distance the muon travels must be at least 3000 m if it is to reach Earth's surface before decaying. The minimum necessary speed for the muon to just reach Earth's surface before decaying can be calculated from V=LrestΔt where Δt=Δtproper1 v2 c2 and Lrest=3000m.

Length contraction in the muon's reference frame:

  L=Lrest1v2c2=3000×1 (0.95c)2c2=937m

Distance traveled by the muon in the muon's reference frame:

  V=LΔt properL=VΔtproper=0.95×3×108×1.56×106=444.6m

Because 444.6 m <937 m, the muon does not make it to Earth.

Time dilation in Earth's reference frame:

  Δt=Δtproper1 v2 c2 =1.56×1061 (0.95c)2 c2 =5×106s

Distance traveled in Earth's reference frame:

  V=L restΔtLrest=VΔt=0.95×3×108×5×106=1425m

Because 1425 m <3000 m, the muon does not make it to Earth.

Conclusion:

The muon does not make it to Earth

To determine

(b)

The minimum speed of the muon so that it just barely reaches Earth's surface after one half-life?

Expert Solution
Check Mark

Answer to Problem 77QAP

The minimum speed = 0.988c

Explanation of Solution

Calculation:

   V= L rest Δt = L rest 1 v 2 c 2 Δ t proper

   V 1 v 2 c 2 = L rest Δ t proper

   ( V c 1 v 2 c 2 ) 2 = ( L rest cΔ t proper ) 2

   v 2 c 2 1 v 2 c 2 = ( L rest cΔ t proper ) 2

   v 2 c 2 = ( L rest cΔ t proper ) 2 1 v 2 c 2

   v 2 c 2 = ( L rest cΔ t proper ) 2 ( L rest cΔ t proper ) 2 v 2 c 2

   v 2 c 2 + ( L rest cΔ t proper ) 2 v 2 c 2 = ( L rest cΔ t proper ) 2

   v 2 c 2 ( 1+ ( L rest cΔ t proper ) 2 )= ( L rest cΔ t proper ) 2 v 2 c 2 = ( L rest cΔ t proper ) 2 ( 1+ ( L rest cΔ t proper ) 2 )

   v c = ( L rest cΔ t proper ) 2 ( 1+ ( L rest cΔ t proper ) 2 ) = ( 3000 3× 10 8 ×1.56× 10 6 ) 2 ( 1+ ( 3000 3× 10 8 ×1.56× 10 6 ) 2 ) =0.988

   v=0.988c

Conclusion:

The minimum speed = 0.988c

Want to see more full solutions like this?

Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!
Students have asked these similar questions
a 500-n block is dragged along a horizontal surface by an applied force t at an angle of 30.0° (see figure). the coefficient of kinetic friction is uk = 0.400 and the block moves at a constant velocity. what is the magnitude of the applied force T in newtons?
a 500-n block is dragged along a horizontal surface by an applied force t at an angle of 30.0° (see figure). the coefficient of kinetic friction is uk = 0.400 and the block moves at a constant velocity. what is the magnitude of the applied force T in newtons?
Block A, with a mass of 10 kg, rests on a 30° incline. The coefficient of kinetic friction is 0.20. The attached string is parallel to the incline and passes over a massless, frictionless pulley at the top. Block B, with a mass of 15.0 kg. is attached to the dangling end of the string. What is the acceleration of Block B in m/s?  show all steps please

Chapter 25 Solutions

COLLEGE PHYSICS,VOLUME 1

Ch. 25 - Prob. 11QAPCh. 25 - Prob. 12QAPCh. 25 - Prob. 13QAPCh. 25 - Prob. 14QAPCh. 25 - Prob. 15QAPCh. 25 - Prob. 16QAPCh. 25 - Prob. 17QAPCh. 25 - Prob. 18QAPCh. 25 - Prob. 19QAPCh. 25 - Prob. 20QAPCh. 25 - Prob. 21QAPCh. 25 - Prob. 22QAPCh. 25 - Prob. 23QAPCh. 25 - Prob. 24QAPCh. 25 - Prob. 25QAPCh. 25 - Prob. 26QAPCh. 25 - Prob. 27QAPCh. 25 - Prob. 28QAPCh. 25 - Prob. 29QAPCh. 25 - Prob. 30QAPCh. 25 - Prob. 31QAPCh. 25 - Prob. 32QAPCh. 25 - Prob. 33QAPCh. 25 - Prob. 34QAPCh. 25 - Prob. 35QAPCh. 25 - Prob. 36QAPCh. 25 - Prob. 37QAPCh. 25 - Prob. 38QAPCh. 25 - Prob. 39QAPCh. 25 - Prob. 40QAPCh. 25 - Prob. 41QAPCh. 25 - Prob. 42QAPCh. 25 - Prob. 43QAPCh. 25 - Prob. 44QAPCh. 25 - Prob. 45QAPCh. 25 - Prob. 46QAPCh. 25 - Prob. 47QAPCh. 25 - Prob. 48QAPCh. 25 - Prob. 49QAPCh. 25 - Prob. 50QAPCh. 25 - Prob. 51QAPCh. 25 - Prob. 52QAPCh. 25 - Prob. 53QAPCh. 25 - Prob. 54QAPCh. 25 - Prob. 55QAPCh. 25 - Prob. 56QAPCh. 25 - Prob. 57QAPCh. 25 - Prob. 58QAPCh. 25 - Prob. 59QAPCh. 25 - Prob. 60QAPCh. 25 - Prob. 61QAPCh. 25 - Prob. 62QAPCh. 25 - Prob. 63QAPCh. 25 - Prob. 64QAPCh. 25 - Prob. 65QAPCh. 25 - Prob. 66QAPCh. 25 - Prob. 67QAPCh. 25 - Prob. 68QAPCh. 25 - Prob. 69QAPCh. 25 - Prob. 70QAPCh. 25 - Prob. 71QAPCh. 25 - Prob. 72QAPCh. 25 - Prob. 73QAPCh. 25 - Prob. 74QAPCh. 25 - Prob. 75QAPCh. 25 - Prob. 76QAPCh. 25 - Prob. 77QAPCh. 25 - Prob. 78QAPCh. 25 - Prob. 79QAPCh. 25 - Prob. 80QAPCh. 25 - Prob. 81QAPCh. 25 - Prob. 82QAPCh. 25 - Prob. 83QAPCh. 25 - Prob. 84QAPCh. 25 - Prob. 85QAPCh. 25 - Prob. 86QAPCh. 25 - Prob. 87QAPCh. 25 - Prob. 88QAPCh. 25 - Prob. 89QAPCh. 25 - Prob. 90QAP
Knowledge Booster
Background pattern image
Physics
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, physics and related others by exploring similar questions and additional content below.
Similar questions
SEE MORE QUESTIONS
Recommended textbooks for you
Text book image
University Physics Volume 3
Physics
ISBN:9781938168185
Author:William Moebs, Jeff Sanny
Publisher:OpenStax
Text book image
Classical Dynamics of Particles and Systems
Physics
ISBN:9780534408961
Author:Stephen T. Thornton, Jerry B. Marion
Publisher:Cengage Learning
Text book image
College Physics
Physics
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Cengage Learning
Text book image
College Physics
Physics
ISBN:9781938168000
Author:Paul Peter Urone, Roger Hinrichs
Publisher:OpenStax College
Text book image
College Physics
Physics
ISBN:9781285737027
Author:Raymond A. Serway, Chris Vuille
Publisher:Cengage Learning
Text book image
Principles of Physics: A Calculus-Based Text
Physics
ISBN:9781133104261
Author:Raymond A. Serway, John W. Jewett
Publisher:Cengage Learning
Time Dilation - Einstein's Theory Of Relativity Explained!; Author: Science ABC;https://www.youtube.com/watch?v=yuD34tEpRFw;License: Standard YouTube License, CC-BY