Physics for Scientists and Engineers: Foundations and Connections
Physics for Scientists and Engineers: Foundations and Connections
1st Edition
ISBN: 9781133939146
Author: Katz, Debora M.
Publisher: Cengage Learning
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Question
Chapter 39, Problem 83PQ

(a)

To determine

The speed of the first piece inspected by an observer in the lab.

(a)

Expert Solution
Check Mark

Answer to Problem 83PQ

The speed of the first piece inspected by an observer in the lab is 0.990c.

Explanation of Solution

Write the expression for the Lorentz constant.

    γ1=11v12c2

Here, Lorentz constant is γ1, speed of the first piece inspected by an observer in the lab is v1 and speed of light is c.

Write the expression for the momentum in fixed frame.

    p1=m1v1

Here, momentum in fixed frame is p1, rest mass is m1 and velocity is v1.

Write the expression for the momentum with respect to laboratory frame.

    p1'=γ1p1

Here, momentum with respect to laboratory frame is p'1.

Substitute m1v1 for p1 and 11v12c2 for γ1 in the above equation to find p1'.

    p1'=(11v12c2)m1v1=m1v11v12c2

Conclusion:

Substitute 1.45MeV/c2 for m1 and 10.0MeV/c for p'1 in the above equation to find v1.

    10.0MeV/c=(1.45MeV/c2)v11v21c2v11v12c2=10.0MeV/c(1.45MeV/c2)=6.896c

Take square both sides.

    (v11v21c2)2=(6.896c)2v121v12c2=47.56c2v12c2v12c2=47.56c2c2v12c2v12=47.56c2

Further solve the above equation.

    c2v12=47.56c2(c2v12)c2v12=47.56c447.56c2v12c2v12+47.56c2v12=47.56c4v12+47.56v12=47.56c2

Further solve the above equation.

    v12(1+47.56)=47.56c2v12(48.56)=47.56c2v12=47.56c248.56v12=0.9794c2

Further solve the above equation.

    v1=0.9794c=0.990c

Therefore, the speed of the first piece inspected by an observer in the lab is 0.990c.

(b)

To determine

The speed of the second piece inspected by an observer in the lab.

(b)

Expert Solution
Check Mark

Answer to Problem 83PQ

The speed of the second piece inspected by an observer in the lab is 0.989c.

Explanation of Solution

Write the expression for the Lorentz constant.

    γ2=11v22c2

Here, Lorentz constant is γ2, speed of the first piece inspected by an observer in the lab is v2 and speed of light is c.

Write the expression for the momentum in fixed frame.

    p2=m2v2

Here, momentum in fixed frame is p2, rest mass is m2 and velocity is v2.

Write the expression for the momentum with respect to laboratory frame.

    p2'=γ2p2

Here, momentum with respect to laboratory frame is p2'.

Substitute m2v2 for p2 and 11v22c2 for γ1 in the above equation to find p2'.

    p2'=(11v22c2)m2v2=m2v21v22c2

Conclusion:

Substitute 0.850MeV/c2 for m2 and 5.75MeV/c for p'2 in the above equation to find v2.

    5.75MeV/c=(0.850MeV/c2)v21v22c2v21v22c2=5.75MeV/c(0.850MeV/c2)=6.764c

Take square both sides.

    (v21v22c2)2=(6.764c)2v221v22c2=45.76c2v22c2v22c2=45.76c2c2v22c2v22=45.76c2

Further solve the above equation

    c2v22=45.76c2(c2v22)c2v22=45.76c445.76c2v22c2v22+45.76c2v22=45.76c4v22+45.76v22=45.76c2

Further solve the above equation

    v22(1+45.76)=45.76c2v22(46.76)=45.76c2v22=45.76c246.76v22=0.9786c2

Further solve the above equation

    v2=0.9786c=0.989c

Therefore, the speed of the second piece inspected by an observer in the lab is 0.989c.

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Chapter 39 Solutions

Physics for Scientists and Engineers: Foundations and Connections

Ch. 39 - Prob. 6PQCh. 39 - Prob. 7PQCh. 39 - Prob. 8PQCh. 39 - Prob. 9PQCh. 39 - Prob. 10PQCh. 39 - Prob. 11PQCh. 39 - Prob. 12PQCh. 39 - Prob. 13PQCh. 39 - Prob. 14PQCh. 39 - Prob. 15PQCh. 39 - Prob. 16PQCh. 39 - Prob. 17PQCh. 39 - Prob. 18PQCh. 39 - Prob. 19PQCh. 39 - Prob. 20PQCh. 39 - Prob. 21PQCh. 39 - Prob. 22PQCh. 39 - Prob. 23PQCh. 39 - A starship is 1025 ly from the Earth when measured...Ch. 39 - A starship is 1025 ly from the Earth when measured...Ch. 39 - Prob. 26PQCh. 39 - Prob. 27PQCh. 39 - Prob. 28PQCh. 39 - Prob. 29PQCh. 39 - Prob. 30PQCh. 39 - Prob. 31PQCh. 39 - Prob. 32PQCh. 39 - Prob. 33PQCh. 39 - Prob. 34PQCh. 39 - Prob. 35PQCh. 39 - Prob. 36PQCh. 39 - Prob. 37PQCh. 39 - Prob. 38PQCh. 39 - As measured in a laboratory reference frame, a...Ch. 39 - Prob. 40PQCh. 39 - Prob. 41PQCh. 39 - Prob. 42PQCh. 39 - Prob. 43PQCh. 39 - Prob. 44PQCh. 39 - Prob. 45PQCh. 39 - Prob. 46PQCh. 39 - Prob. 47PQCh. 39 - Prob. 48PQCh. 39 - Prob. 49PQCh. 39 - Prob. 50PQCh. 39 - Prob. 51PQCh. 39 - Prob. 52PQCh. 39 - Prob. 53PQCh. 39 - Prob. 54PQCh. 39 - Prob. 55PQCh. 39 - Prob. 56PQCh. 39 - Consider an electron moving with speed 0.980c. a....Ch. 39 - Prob. 58PQCh. 39 - Prob. 59PQCh. 39 - Prob. 60PQCh. 39 - Prob. 61PQCh. 39 - Prob. 62PQCh. 39 - Prob. 63PQCh. 39 - Prob. 64PQCh. 39 - Prob. 65PQCh. 39 - Prob. 66PQCh. 39 - Prob. 67PQCh. 39 - Prob. 68PQCh. 39 - Prob. 69PQCh. 39 - Prob. 70PQCh. 39 - Joe and Moe are twins. In the laboratory frame at...Ch. 39 - Prob. 72PQCh. 39 - Prob. 73PQCh. 39 - Prob. 74PQCh. 39 - Prob. 75PQCh. 39 - Prob. 76PQCh. 39 - Prob. 77PQCh. 39 - In December 2012, researchers announced the...Ch. 39 - Prob. 79PQCh. 39 - Prob. 80PQCh. 39 - How much work is required to increase the speed of...Ch. 39 - Prob. 82PQCh. 39 - Prob. 83PQCh. 39 - Prob. 84PQCh. 39 - Prob. 85PQ
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