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```					Universal Speed Limit
and Relativistic Mass
Physics 12
Length Contraction
A concept related to time dilation is that of
length contraction
In a way that is similar to time changes
depending on the frame of reference,
length is also affected
This can be used to explain the behaviour
of the mu meson (or muon)
Muon
Muons are particles that are created high
in the atmosphere due to UV radiation
These particles are created about 9000m
above the surface of the Earth and travel
Muon
t0
t 
Time Dilation                          v2
   According to time                1 2
dilation, the muon’s               c
half-life (measured in             2 s
the frame of reference    t 
of the Earth) should be              (.998c) 2
30μs in the muon’s              1
frame
c2
   As a result, the muon     t  30 s
can travel a .998c for    d  vt
30μs covering a
distance of 9000m         d  .998c(30s )
d  9000m
Muon
Length Contraction                        v  2

   According to length         L  L0 1  2
contraction, the distance
c
that the muon needs to
(.998c) 2
move through (measured      L  9000m 1 
in the frame of reference                    c2
of the Earth) should be
L  600m
600m in the muon’s frame
   As a result, the Earth      d  vt
rushes towards the muon
at .998c for 2μs covering
d  .998c(2 s )
a distance of 600m          d  600m
Practice Problems
Page 824
   4-6
Gamma
1

For calculations using
special relativity,
2
gamma is often used               v
in order to save time         1 2
when writing                      c
equations:
t  t0
L0
L

Universal Speed Limit
When we consider gamma, we know that
we must be dealing with real numbers so
the value under the root must be positive
Therefore speed (v) cannot be greater
than or equal to the speed of light (c) or
the denominator becomes imaginary or
zero
This speed limit only applies to objects
with mass (therefore the massless photon
can travel at the speed of light)
Cerenkov’s Glow
While it is impossible for anything to travel
faster than light in a vacuum, it is possible
for an object to travel faster than light in a
medium
which is seen in the cooling pools of a
nuclear power plant
The particles in the water are travelling
faster than the speed of light in the water
and the glow is thus produced
Mass and Energy
While the gamma term leads to the
mathematical understanding that the
speed of light is the limit for massive
objects, it does not explain why
The reason is found in Newton’s Second
Law and Einstein’s Special Theory of
Relativity
Einstein found that in addition to time
dilation and length contraction, mass is
also affected by relativistic effects
Relativistic Mass
As a result, the mass
increases as an
object’s speed                m0
increases                m
2
Therefore, according             v
to Newton, an object
1 2
c
travelling at .999c
would require an         m  m0
infinite force to
accelerate it to c as:
Practice Problems
Page 825
   1-8
Page 830
   7-9

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