special relativity: time dilation and length contraction sph4u
TRANSCRIPT
Special Relativity: Time Dilation and Length Contraction
SPH4U
Review Concepts
• A child stands in the aisle of a train moving eastward at 40 m/s. For a short time, the child runs forward in the same direction as the train is moving at a constant speed, covering 15 m in 3 s.
(a) Calculate the velocity of the child relative to an observer on the train.
(b) Calculate the velocity of the child relative to an observer on the ground.
Review Concepts
• A child stands in the aisle of a train moving eastward at 40 m/s. For a short time, the child runs forward in the same direction as the train is moving at a constant speed, covering 15 m in 3 s.
(c) A second train moves westward at a speed of 30 m/s. Calculate the velocity of the child relative to an observer on the second train.
Review Concepts
• For each situation, identify whether the person is in an inertial or a non-inertial frame of reference. Explain your choices.
(a)a taxi driver accelerating away from a stop light(b) a child riding around the outer edge of a carousel(c) a race car travelling at 200 km/h down a straight
section of track(e) a student riding swiftly down the first hill of a roller
coaster
Skills Review
• Use the wave equation to calculate the frequency of each of the wavelengths below.
(a) 620 nm(b) 540 nm(c) 320 nm
Skills Review
• Which wavelength from the previous question has the highest energy?
• (a) 640 nm (4.7 x 1014 Hz)• (b) 510 nm (5.9 x 1014 Hz)• (c) 320 nm (9.4 x 1014 Hz)• (c) 320 nm (E = hf or hc/ƛ)• NOTE!• The smaller the wavelength the more energy it has. Recall that radio
waves have a long wavelength (and thus little energy) so they do not hurt us. However, UV rays have a much shorter wavelength (and thus more energy) and they do hurt us (i.e. sunburns).
A rocket ship is heading toward you at ½ c. How fast does the light from its headlamps travel toward you?
a) cb) 1.5 cc) 0.5 cd) something else
The speed of light is c in all frames.
Twin Paradox
• One twin (Al) stays on earth and the other twin (Bert) goes off into space very, very fast. From the viewpoint (frame) of Al, Bert ages slower. When Bert returns, he is younger than Al.
• But for Bert, it is the earth that is moving. So Al should age slower than himself, and be younger when they meet on Earth.
• But who is right?
You have been placed in a rocket traveling at 99 % c and cannot look out the window. List all the tests that you could do to tell that you are traveling so fast.
There are no tests that you can do. All inertial frames are equivalent.
Galilean Relativity
Relative motion:Two objects are in relative motion if they are moving at different velocities.Each object has its own reference frame.– Example, two kids playing ball on a moving train car: What
are some ways to describe their motion?
Ball w.r.t kid, kid w.r.t train, train w.r.t earth, earth w.r.t sun,
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
• Thought Experiment You are on your Prom somewhere overlooking Lake Ontario when fireworks begin. Two fireworks ignite at exactly the same time..one off to your left & another to your right. About 500m into the Lake, a speedboat is travelling at 80km/hr. Do the speedboat passengers see the fireworks igniting simultaneously??
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
