Physics for Scientists and Engineers: Foundations and Connections
Physics for Scientists and Engineers: Foundations and Connections
1st Edition
ISBN: 9781337026345
Author: Katz
Publisher: Cengage
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Chapter 39, Problem 40PQ
To determine

The proof of vy=vyγ(1+vrelc2vx) and vy=vyγ(1vrelc2vx).

Expert Solution & Answer
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Answer to Problem 40PQ

The proof of vy=vyγ(1+vrelc2vx) and vy=vyγ(1vrelc2vx) has been determined.

Explanation of Solution

Write the Lorentz’s transformation equations.

  y=y                                                                                                                          (I)

  t=γ(t+vrelxc2)                                                                                                        (II)

  t=γ(tvrelxc2)                                                                                                       (III)

Here, y,x and t are the position and time in prime coordinate, γ is relativistic factor,y, x and t are the position and time in unprimed coordinate, vrel is the relative velocity between two coordinates and c is the speed of light.

Differentiate above Lorentz’s equations to find dy, dt and dt.

  dy=dy                                                                                                                  (IV)

  dt=γ(dt+vreldxc2)                                                                                                (V)

  dt=γ(dtvreldxc2)                                                                                                 (VI)

Case 1:

Write the equation for the velocity of the particle in the unprimed frame.

  vy=dydt                                                                                                                   (VII)

Substitute equation (IV) and equation (V) in above equation to find vy.

  vy=dyγ(dt+vreldxc2)=dyγdt(1+vrelc2dxdt)=dydtγ(1+vrelc2dxdt)                                                                                          (VIII)

Write the equation for the velocity of the particle in primed frame along y direction.

  vy=dydt                                                                                                                  (IX)

Write the equation for the velocity of the particle in primed frame along x direction.

  vx=dxdt                                                                                                                (X)

Substitute equation (IX) and (X) in equation (VIII) to find vy.

  vy=vyγ(1+vrelc2vx)                                                                                                  (XI)

Case 2:

Write the equation for the velocity of the particle in the primed frame.

  vy=dydt                                                                                                               (XII)

Substitute dy for dy and γ(dtvreldxc2) for dt in above equation to find vy.

  vy=dyγ(dtvreldxc2)=dyγdt(1vrelc2dxdt)=dydtγ(1vrelc2dxdt)                                                                                          (XIII)

Write the equation for the velocity of the particle in unprimed frame along y direction.

  vy=dydt                                                                                                                (XIV)

Write the equation for the velocity of the particle in unprimed frame along x direction.

  vx=dxdt                                                                                                                (XV)

Substitute equation (IX) and (X) in equation (VIII) to find vy.

  vy=vyγ(1vrelc2vx)                                                                                                 (XVI)

Conclusion:

Therefore, the transformation of a velocity component perpendicular to vrel is vy=vyγ(1+vrelc2vx) and the inverse transformation from laboratory frame to the primed frame is vy=vyγ(1vrelc2vx).

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Chapter 39 Solutions

Physics for Scientists and Engineers: Foundations and Connections

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