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Chapter 39, Problem 14PQ

(a)

To determine

The proof for the expression t=γ(t'+vrelc2x').

(a)

Expert Solution
Check Mark

Answer to Problem 14PQ

The Lorentz transformation relating x and x' confirms the equation t=γ(t'+vrelc2x').

Explanation of Solution

Write the expression to obtain the position of the object along x axis in primed frame.

    x'=γ(xvrelt)                                                                                                   (I)

Here, x is the position of the object along x axis in laboratory frame, γ is the Lorentz factor, x' is the position of the object along x axis in primed frame, vrel is the relative velocity of the object and t is the time interval in laboratory frame.

Write the expression to obtain the position of the object along x axis in laboratory frame.

    x=γ(x'+vrelt')                                                                                                    (II)

Here, x is the position of the object along x axis in laboratory frame, γ is the Lorentz factor, x' is the position of the object along x axis in primed frame, vrel is the relative velocity of the object and t' is the time interval in primed frame.

Substitute γ(x'+vrelt') for x in equation (I).

    x'=γ((γ(x'+vrelt'))vrelt)x'=γ2(x'+vrelt')γvreltx'=γ2x'+γ2vrelt'γvreltγvrelt=γ2x'+γ2vrelt'x'

Further solve the above equation.

    t=γ21γvrelx'+γt'                                                                                                 (III)

Write the expression to obtain the Lorentz factor.

    γ=11(vrelc)2

Here, γ is the Lorentz factor, vrel is the relative velocity of the object and c is the speed of light.

Take square both sides in the above equation.

    (γ)2=(11(vrelc)2)2

Subtract 1 from both sides in the above equation.

    (γ)21=11(vrelc)21=(vrelc)21(vrelc)2

Divide both sides by γ in the above equation.

    (γ)21γ=[(vrelc)21(vrelc)2]γ

Substitute 11(vrelc)2 for γ in the above equation.

    (γ)21γ=[(vrelc)21(vrelc)2][11(vrelc)2]=1(vrelc)2(vrelc)21(vrelc)2=(vrelc)21(vrelc)2

Substitute 11(vrelc)2 for γ in the above equation.

    (γ)21γ=γ(vrelc)2                                                                                                (IV)

Substitute γ(vrelc)2 for (γ)21γ in equation (III).

    t=γ(vrelc)2x'vrel+γt'=γ(t'+vrelc2x')

Therefore, the Lorentz transformation relating x and x' confirms the equation t=γ(t'+vrelc2x').

(b)

To determine

The Lorentz transformation relating x and x' confirms the equation t'=γ(tvrelc2x).

(b)

Expert Solution
Check Mark

Answer to Problem 14PQ

The Lorentz transformation relating x and x' confirms the equation t'=γ(tvrelc2x).

Explanation of Solution

Rewrite (II).

  x=γ(x'+vrelt')

Substitute γ(x+vrelt) for x' in equation (II).

    x=γ((γ(x+vrelt))+vrelt')x=γ2(x+vrelt)+γvrelt'x=γ2x+γ2vrelt+γvrelt'γvrelt'=x(γ2x+γ2vrelt)

Further solve the above equation.

    t'=γ21γvrelxγt                                                                                                  (V)

Consider equation (IV).

    (γ)21γ=γ(vrelc)2

Substitute γ(vrelc)2 for (γ)21γ in equation (V).

    t'=γ(vrelc)2xvrel+γt=γ(tvrelc2x)

Therefore, the Lorentz transformation relating x and x' confirms the equation t'=γ(tvrelc2x).

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Chapter 39 Solutions

Webassign Printed Access Card For Katz's Physics For Scientists And Engineers: Foundations And Connections, 1st Edition, Single-term

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