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Lorentz Velocity Transformation Equations |
Consider
two coordinate systems in standard configuration observing a particle of
relativistic velocity. Let the particle’s velocity in system S be 
and in system S’ be 
. Then the classical relationship between the components of
these velocities will be given by
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The
Lorentz Velocity Transformation Equations |
Transform |
Inverse
Transform |
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ux,
uy, uz = components of velocity in rest frame S u’x,
u’y, u’z = components of velocity in moving frame S’ v = velocity of S’ along
+x-axis of S g = g(v) = gamma for v = (1-v2/c2)-1/2 |
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These
equations follow immediately from the derivatives of the Lorentz coordinate
transformation equations. As usual, the inverse transformation equations
follow from the transformation equations by exchanging the primes with the
unprimes and replacing v with –v.
One
of the more interesting consequences of relativistic velocity addition is the
headlight effect that causes fast moving objects to beam their radiant energy
forward in the direction they are moving. See Serway pp. 28-29 and Tipler
pp. 24-25 for further discussions, derivations, and examples of the application
of these equations.
Problems:
P1. Symbol Conversion
Show
that the Lorentz velocity equation given in the previous page for vAC
follows immediately from the first inverse Lorentz velocity equation given
above for ux through a simple substitution of appropriate
symbols.
P2. Derivation
Derive
the above Lorentz velocity transformation equations by differentiating the
Lorentz coordinate transformation equations and solving simultaneously for the
velocity components in terms of 
and 
, etc. (b) Also derive the inverse transforms by the proper
substitution of variables.
P3. Vector Velocities
Two
spaceships pass one another in space. Relative to the earth, the first is
moving “northward” at 0.6 c and the second is moving “westward” at 0.8 c. What
is the magnitude and direction of the velocity of the first spaceship with
respect to the second?
P4. Photon Forward
A
beam of light is aimed at an angle of 45 o out the back window of a
spaceship traveling at 0.99 c. (So the angle between the beam and the front of
the spaceship is 135o.) (a) Use the velocity transformation equation
to find the magnitude and direction of the velocity of the beam of light
relative to an observer at rest? (b) If the wavelength of the light emitted
from the spaceship is 500 nm (green), then what is the wavelength of the light
seen by the observer at rest? (Hint: use the relativistic transverse Doppler
equation for part b.)
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ModPhy1/Unit1/SpecialRelativity/RelativeView/Velocity/