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Velocity is Relative |
The Galilean
Velocity Transformation Equation
Velocity
is a relative quantity in both classical and relativistic physics. But
classical physics admits the possibility of an absolute rest frame. According
to classical physics,
The
classical equations for transforming relative velocities from one inertial
reference frame to another are called the Galilean velocity
transformation equations. When reduced to linear form, they become:
|
vAB = velocity of A relative
to B, etc. vBA = -vAB |
vAC = vAB +
vBC |
According
to this equation, relative velocities simply add to one another. For this
reason, velocity transformation equations are also called velocity addition
equations.
As
an illustration of the application of this equation, suppose two cosmic rays
approach the earth from opposite directions at 0.9 c. Then v1E
= 0.9 c, v2E = -0.9 c = - vE2
, and v12 = v1E + vE2 = 0.9 c + 0.9 c = 1.8 c. Therefore,
according to classical physics, the velocity of one cosmic ray with respect to
the other is greater than the speed of light.
The Lorentz
Velocity Transformation Equation
Relativistic
physics denies the existence of an absolute rest frame. Therefore, relativity
always treats velocity as a relative quantity, never as an absolute quantity.
The
relativistic velocity addition equations are called the Lorentz velocity transformation
equations. When reduced to linear form, they become:
|
vAB = velocity of A relative
to B, etc. vBA = -vAB |
|
As
you can see, the relativistic velocity addition equation contains an additional
factor in the denominator of the expression that prevents the sum of two
velocities from ever exceeding the speed of light.
For
example, consider the same two cosmic rays treated classically above and
substitute their relative velocities into the Lorentz velocity transformation
equation. Then, v12 = (0.9c+0.9c)/(1+(0.9c)(0.9c)/c2)
= (1.8c)/(1.81) = 0.9945 c. Therefore, according to relativity, the velocity of
the first cosmic ray with respect to the second is close to, but still less
than, the speed of light.
Questions:
Q1. An object A is moving in the +y-direction with a speed of 0.8
c. Object B is moving in the -x-direction with a speed of 0.8 c. What is the
speed of object A with respect to object B according to relativity.
Problems:
P1. Classical Boost Sequence
A
large acceleration over a short period of time is called a “boost.” (a) Use the
Galilean velocity addition equations to determine the speed of a spaceship
boosted twice, each time to half the speed of light. (b) What is its speed
after the third such boost? (c) After the fourth? (d) After the fifth? (e)
After the n’th? (f) Will such a process of multiple boosts ever accelerate the
spaceship faster than the speed of light?
P2. Relativistic Boost Sequence
Use
the Lorentz velocity addition equations to determine the speed of a spaceship
twice boosted to half the speed of light. (b) What is its speed after the third
such boost? (c) After the fourth? (d) After the fifth? (e) After the n’th? (f)
Will such a process of multiple boosts ever accelerate the spaceship faster
than the speed of light?
P3. Correspondence Principle
Prove
that the Lorentz velocity addition equation reduces to the Galilean velocity
addition equation for small velocities.
P4. Velocity of a Photon
A
photon of speed c passes a spaceship of speed v moving in the same direction
relative to the earth. (a) Use the relativistic velocity addition equation to
show that the speed of the photon relative to the spaceship is also c. (a) Show
that the same is true for a photon traveling in the opposite direction.
P4. A Photon Approaching a Photon
(a)
Use the velocity addition equation to determine the relative velocity of two
photons traveling in opposite directions. (b) Use the velocity addition equation
to determine the relative velocity of two photons traveling in the same
direction.
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ModPhy1/Unit1/SpecialRelativity/RelativeView/