Physics for Scientists and Engineers
Physics for Scientists and Engineers
6th Edition
ISBN: 9781429281843
Author: Tipler
Publisher: MAC HIGHER
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Chapter 39, Problem 55P
To determine

To Show:The momentum and energy of a particle in frame S’ are related to its momentum and energy in frame S by the transformation equations.

Expert Solution & Answer
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Answer to Problem 55P

  px=γ(pxvEc2)

  p'y=py

  p'z=pz

  E'c=γ(Ecvpxc) .

Explanation of Solution

Given:

A particle moving with speed u along the y-axis in frame S .

Formula Used:

Relativistic momentum, p=mu1u2c2

Relativistic energy, E=mc21u2c2

Where, u is the velocity of the particle and u is its speed.

Calculations:

In any inertial frame

  p=mu1u2c2

As the particle is moving in the y axis

  px=pz=0

  py=muy1u2c2

Substitute these values in the transformation equations:

  p'x=γ(0vEc2)=γvEc2

  p'y=py

  p'z=0

  E'c=γ(Ec0)=γEc

In S’ frame, the momentum components are

  px'=mux'1u'2c2

  py'=muy'1u'2c2

  pz'=muz'1u'2c2

The inverse velocity transformations are

  ux'=uxv1uxvc2

  uy'=uy1uyvc2

  uz'=uz1uzvc2

Substitute

  ux=uz=0

  uy=u

Thus,

  ux'=v

  uy'=γu and

  uz'=0

  u'2=v2+u2γ2

To verify that

  p'z=pz=0

  pz'=m(0)1u'2c2=0

Next

  p'y=py

  py'=muy'1u'2c2=muγ1v2c2u2γ2c2=mu1u'2c21u'2c2γ1v2c2u2γ2c2

  py'=mu1u2c2(1u2c2)(1v2c2)1v2c2u2c2(1v2c2)=py1v2c2u2c2(1v2c2)1v2c2u2c2(1v2c2)

Hence p'y=py

Next, to verify px=γ(pxvEc2)

  px'=mux'1u'2c2=mvγ1v2c2u2γ2c2=γvc2mc21u2c2γ11u2c21v2c2u2γ2c2

  px'=γvc2E(1u2c2)(1v2c2)1v2c2u2c2(1v2c2)=γvc2E1v2c2u2c2(1v2c2)1v2c2u2c2(1v2c2)

  px'=γvc2E

Finally

  E'c=γ(Ecvpxc)=γEc,or E'=γE

  E'=mc21u'2c2=γmc21u2c2γ11u2c21u'2c2=γEγ11u2c21v2c2u2γ2c2

  E'=γE(1u2c2)(1v2c2)1v2c2u2c2(1v2c2)=γE1v2c2u2c2(1v2c2)1v2c2u2c2(1v2c2)

  E'=γE

The x,y,z and t transformation equations are

  x'=γ(xvt),y'=y,z'=z and t'=γ(tvxc2)

The x,y,z and ct transformation equations are

  x'=γ(xvcct),y'=y,z'=zand ct'=γ(ctvcx)

The px,py,pz and E/c transformation equations are

  p'x=γ(pxvcEc),p'y=py,p'z=pz and E'c=γ(Ecvcpx)

Conclusion:

  px=γ(pxvEc2)

  p'y=py

  p'z=pz

  E'c=γ(Ecvpxc) .

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