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• the Lorentz transformation equations
• Once again Ś is our frame
moving at a speed v relative
to S.
8.The Lorentz Velocity TransformationEquations
Suppose two observers in relative motion with
respect to each other are both observing the
motion of an object. Previously, we defined an
event as occurring at an instant of time. Now, we
wish to interpret the “event” as the motion of the
object. We know that the Galilean velocity
transformation is valid for low speeds. Which is
A particle moves through a displacement of
magnitude dx along the x axis in a time interval dt
as measured by an observer in S. the corresponding
displacement dx́ measured by an observer in Ś is
• where frame Ś is moving with speed v in the x
direction relative to frame S. Because dt= dt́, we
find that
• where ux and úx are the x components of the
velocity of the particle measured by observers in
S and Ś, respectively.
• We use the symbol u for particle velocity rather
than v, which is used for the relative velocity of
two reference frames.)
• This is the Galilean velocity transformation
equation.
• How do the observers’ measurements of the
velocity of the object relate to each other if the
speed of the object is close to that of light?
Once again Ś is our frame moving at a speed v
relative to S.
• Suppose that an object has a velocity
component úx measured in the Ś frame, where
• Using Lorentz transformation equations
• We have
• Substituting these values into Equation
• We get
• But dx/dt is just the velocity component ux of
the object measured by an observer in S, and so
this expression becomes
• If the object has velocity components along the
y and z axes, the components as measured by
an observer in Ś are
• Note that úy and úz do not contain the
parameter v in the numerator because the
relative velocity is along the x axis.
• When v is much smaller than c (the
nonrelativistic case), the denominator of
Equation 39.16 approaches unity, and so
úx ≈ ux - v, which is the Galilean velocity
transformation equation.
• In another extreme, when ux = c, Equation 39.16
becomes
• From this result, we see that a speed measured
as c by an observer in S is also measured as c by
an observer in Ś
independent of the relative motion of S and Ś. Note
that this conclusion is consistent with Einstein’s
second postulate—that the speed of light must
be c relative to all inertial reference frames.
Furthermore, we find that the speed of an object
can never be measured as larger than c. That is,
the speed of light is the ultimate speed. We
return to this point later.
Example:
Two spacecraft A and B are moving in opposite directions, as
shown in Figure 39.15. An observer on the Earth measures
the speed of craft A to be 0.750c and the speed of craft B
to be 0.850c. Find the velocity of craft B as observed by the
crew on craft A.
To finalize this problem, note that the negative sign indicates
that craft B is moving in the negative x direction as
observed by the crew on craft A. Note that the speed is less
than c. That is, an object whose speed is less than c in one
frame of reference must have a speed less than c in any
other frame.
• Example:
• A Klingon spacecraft moves away from the Earth at a
speed of 0.800c (Fig. P39.26). The starship Enterprise
pursues at a speed of 0.900c relative to the Earth.
Observers on the Earth see the Enterprise overtaking
the Klingon craft at a relative speed of 0.100c. With
what speed is the Enterprise overtaking the Klingon
craft as seen by the crew of the Enterprise?
• Solution:
• Example:
Two jets of material from the centre of a radio
galaxy are ejected in opposite directions. Both
jets move at 0.750c relative to the galaxy.
Determine the speed of one jet relative to the
other.
• Solution:
• Example:
• A spacecraft is launched from the surface of the
Earth with a velocity of 0.600c at an angle of
50.0° above the horizontal positive x axis.
Another spacecraft is moving past, with a
velocity of 0.700c in the negative x direction.
Determine the magnitude and direction of the
velocity of the first spacecraft as measured by
the pilot of the second spacecraft.
• Solution:
9.RELATIVISTIC MOMENTUM
Properly describing the motion of particles within
the framework of special relativity requires
generalizing Newton’s laws of motion and the
definitions of momentum and energy. These
generalized definitions reduce to the classical
(nonrelativistic) definitions when v is much less
than c.
First, recall that conservation of momentum
states that when two objects collide, the total
momentum of the system remains constant,
assuming that the objects are isolated, reacting
only with each other
• In order to have momentum conservation in all
inertial frames—even those moving at an
appreciable fraction of c—the definition of
momentum must be modified to read
• where v is the speed of the particle and m is its
mass as measured by an observer at rest with
respect to the particle. Note that when v is much
less than c, the denominator of Equation 26.10
approaches one, so that p approaches mv.
Therefore, the relativistic equation for momentum
reduces to the classical expression when v is small
compared with c.
• The relativistic force F acting on a particle whose linear
momentum is p is defined as
• But
and
• Where mo is the mass of the particle at rest and m is
the particle mass after motion
• then
• This expression is reasonable because it preserves
classical mechanics in the limit of low velocities and is
consistent with conservation of linear momentum for an
isolated system (F= 0) both relativistically and classically.
• under relativistic conditions, the acceleration a of a
particle decreases under the action of a constant
force, in which case a α (1 - u2/c2)3/2. From this
proportionality, we see that as the particle’s speed
approaches c, the acceleration caused by any finite
force approaches zero.
• The mass of the particle will approach infinity
For v=c, the dominator will reach zero which mean that
the mass will be infinity. Which is impossible physical
condition.
• Hence, it is impossible to accelerate a particle from
rest to a speed u≥c.
• This argument shows that the speed of light is the
ultimate speed, as noted at the end of the preceding
section.
EXAMPLE
An electron, which has a mass of 9.11 x 10-31 kg,
moves with a speed of 0.750c. Find the classical
(nonrelativistic) momentum and compare it to
its relativistic counterpart prel.
Solution
First, compute the classical (nonrelativistic)
momentum with v = 0.750c
Multiply this result by γ to obtain the relativistic
momentum:
EXAMPLE
An unstable particle at rest breaks up into two
fragments of unequal mass. The mass of the lighter
fragment is 2.50 x 10-28 kg, and that of the heavier
fragment is 1.67x 10-27 kg. If the lighter fragment
has a speed of 0.893c after the breakup, what is
the speed of the heavier fragment?
Solution
Relativistic momentum of the system of fragments
must be conserved. For total momentum to be zero
after as it was before, we must have, with subscript
2 referring to the heavier fragment, and subscript 1
to the lighter, p2 = p1
10.RELATIVISTIC ADDITION OF VELOCITIES
Imagine of a motorcycle
rider moving with a
speed of 0.80c past a
stationary observer, as
shown in Figure 26.12.
If the rider tosses a ball
in the forward direction
with a speed of 0.70c
relative to himself, what
is the speed of the ball
as seen by the
stationary observer at
the side of the road?
• Common sense and the ideas of Newtonian relativity
say that the speed should be the sum of the two
speeds, or 1.50c. This answer must be incorrect
because it contradicts the assertion that no material
object can travel faster than the speed of light.
• Let two frames or reference be labelled b and d, and
suppose that frame d is moving at velocity vdb in the
position x-direction relative frame b(the velocity with
respect to stationary observer). If the velocity of an
object a as measured in frame d is called vad (The
velocity of the trajectory with respect to the moving
reference)then the velocity of a as measured in
frame b, vab, is given by
• We apply Equation
26.11 to Figure 26.13,
which shows a
motorcyclist, his ball,
and a stationary
observer. We are given
• and we want to find
• v bo = the velocity of the
ball with respect to the
stationary observer
• Thus,