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Special Relativity: Time Dilation and Length Contraction. SPH4U. Einstein’s Postulates. All laws of physics are valid in all inertial (non-accelerated) reference frames. Einstein’s Postulates. Light travels at c = 3.0 x 10 8 m/s relative to all reference frames. Simultaneity. - PowerPoint PPT Presentation
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Special Relativity: Time Dilation and Length Contraction
SPH4U
Einstein’s Postulates
• All laws of physics are valid in all inertial (non-accelerated) reference frames.
Einstein’s Postulates
• Light travels at c = 3.0 x 108 m/s relative to all reference frames.
Simultaneity
This implies that events that are simultaneous (occurring at the same time) for one observer are not simultaneous for an observer in a different reference frame.
C will see the lightning flashes at A and B at the same time.
D will see the flash at B happen before the flash at A.
Measurement
And that observers in different reference frames measure time and length differently.
One observer bounces a laser off a mirror on the ceiling.
The other sees the light travelling a longer path in a longer time.
Time DilationBy the Pythagorean Theorem:
ct ct’
vt’
Time DilationBy the Pythagorean Theorem:
ct ct’
vt’
2
222
22
222
222
22
222222
222
1'
''
''
''
''
c
vtt
tc
vtt
ttc
vt
tctvtc
ctvtct
Time Dilation
(for an event that occurs in the non-prime reference frame)
ct ct’
vt’
2
22
2
1
'1'
cv
ttor
c
vtt
Time Dilation
(for an event that occurs in the non-prime reference frame)
ct ct’
vt’
2
22
2
1
'1'
cv
ttor
c
vtt
dilation = expansion
Gamma
The factor is often written as :2
2
1
1
cv
ttort
t
''
Note that is dimensionless (it has no units).
Length Contraction
Similarly, it can be shown that lengths parallel to the direction of motion are contracted:
2
22
2
1
'1'
cv
LLor
c
vLL
'' LLorL
L
Muons
Muons created in the upper atmosphere have a lifetime of 2.2 s in their reference frame but reach the Earth’s surface because they see the distance to the surface as contracted and/or we see the lifetime as dilated.
Example
What is the apparent lifetime of a muon travelling at 0.99c?
Example
What is the apparent lifetime of a muon travelling at 0.99c?
09.7)99.0(
1
1
1
1
2
2
2
2
cc
cv
Example
What is the apparent lifetime of a muon travelling at 0.99c?
09.7)99.0(
1
1
1
1
2
2
2
2
cc
cv
st
st
tt
16'
)2.2)(09.7('
'
More Practice
Textbook Questions• p. 573 #2, 3• p. 576 #6, 7, 8
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