COLLEGE PHYSICS
COLLEGE PHYSICS
2nd Edition
ISBN: 9781464196393
Author: Freedman
Publisher: MAC HIGHER
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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

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Chapter 25 Solutions

COLLEGE PHYSICS

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
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