Physics
y
132: Lecture 24
Elements of Physics II
A
Agenda
d ffor Today
T d
• Special Theory of relativity
– Inertial vs. nonnon-inertial reference
frames
– Postulates of SR
• Consequences of SR
– Time dilation
– Length contraction
Postulates of Special
Relativity
1) The
Th llaws off Ph
Physics
i have
h
the
th same fform
in all inertial reference frames. (Moving with
constant velocity)
1B) The speed of light (in a vacuum) is always
c,, independent
p
of the observer.
Time dilation
t 

1-
v2
c2
 = proper time

ti
(ti
(time
measured
d iin th
the fframe where
h
th
the
events happen at the same place)
t = dilated time interval (time measured in the frame moving
with respect to the events)
v is relative velocity of objects
c is speed of light
Proper time
• Proper time is the time measured in the
frame where the events happen
pp at the same
place
• It is not the REAL time
time, there is no REAL
time
• All observers measure a ‘correct’
correct time
Give it a try:
Alan and Beth operate synchronized clocks at two
points on the earth’s surface, A and B. Clarissa rides
a train at a constant speed bet
between
een the ttwo
o points A
and B. Which registers the greatest time interval
between the events: Clarissa’s watch, or the
difference between Alan and Beth’s clocks? Assume
these people have super accurate clocks.
(a) Alan and Beth
(b) Clarissa
(c) They will be exactly the same
Length Contraction
 C
Consider
id a rocket
k t which
hi h ttravels
l ffrom th
the sun tto S
Saturn
t
at speed v.
 In the reference frame S of the solar system
system, the
distance traveled is L, and v  L/t, where t is the
time for the rocket to make the journey in the S frame.
Length Contraction
Length Contraction
 In the reference frame S, the rocket is at rest and the
solar system moves to the left at speed v.
 The relative speed between S and S is the same for
both reference frames:
Length Contraction
 The time interval t
t measured in frame S
S is the proper
time  because both events occur at the same position
in frame Sand can be measured by one clock.
 Since the speed is the same in both frames:
 The t cancels,, and the distance L in frame S is:
 where is the proper length measured in the frame
where the objects are at rest.
 The length of any object is less when it is measured in a
reference frame in which the object is moving.
Give it a try:
Alan and Beth operate synchronized clocks at two
points on the earth’s surface, A and B. Clarissa rides
a train at a constant speed bet
between
een the ttwo
o points A
and B. Who measures the proper length of the train?
(a) Alan and Beth
(b) Clarissa
(c) They will all measure the proper length of the train
Clicker Question 4:
• In my spaceship of proper length 5 m I fly
past you at a speed of 0.8c. What do you
measure the length of my ship to be?
(a)
(b)
( )
(c)
(d)
(e)
3.33 m
9m
8 33 m
8.33
4m
3m
Clicker Question 4:
• In my spaceship of proper length 5 m I fly past you at
a speed
d off 0.8c.
0 8 What
Wh do
d you measure the
h llength
h off
my ship to be?
L  L0 1 
L=3m
2
v
c2
 (5m) 1 
(.8c) 2
c2
The Lorentz Transformations
 The spacetime coordinates of an event are measured in
inertial reference frames S and S.
 We wish to transform (x
(x, y,
y z,
z t) to (x,
(x y,
y z,
z t) and vice
versa.
The Lorentz Transformations
 Frame S moves at speed v relative to S.
 The motion is parallel to the x and xaxes.
 We define t  0 and t 0 as the instant when the
origins of S and Scoincide.
 The
Th Lorentz
L
t Transformations
T
f
ti
are:
where
Momentum
• Other things will change as well. As
you can imagine if length and time
change so will other quantities.
p
p is momentum
m0 v
1
m0 is rest mass
2
v
2
c
v is relative velocity of objects
c is speed
p
of light
g
Momentum
• Plot of the new term
added to the
momentum. Term is
more significant for
large v.
p
m0v
1
1
2
v
2
c
Energy and mass
• Mass is equivalent to energy!!!
E tot 
m 0c
2
1
v2
c2
E = m0c2
Energy when at rest (rest energy)
Energy and mass
• Etot = m0c2 +KE
KE 
m 0c
2
1
v2
c2
 m 0c
2
Energy and mass
• Nuclear reactions ((E = mc2)
• Fusion (Sun)
– 2 atoms combine to make a third
– mass of result is less than the first two
combined
• Fission (Atomic bomb)
– 1 atom
t
splits
lit tto make
k ttwo atoms
t
– The mass of the two resultants is less than
the original atom
Nuclear Fission
 Probably the most well
well-known
known
application of the
conservation of total energy is
nuclear
l
fission.
fi i
 A 235U nucleus absorbs a
neutron and then quickly
fragments into two smaller
nuclei and several extra
neutrons.
t
 Mass has been lost and
converted to an equivalent
amount of kinetic energy in
the fission products.
 This generates heat.
Cosmic Speed Limit
• Unless you’re on Star Trek nothing can
move faster then the speed of light
– Tachyon
vBG
vBT  vTG

vBT vTG
1
2
c
t 
t 0
1-
v2
c2
Finally
y
• Special relativity has been tested many
titimes
– Never has it been violated
• More general form of Newton;s and others
equations
For v << c
p
m0v
1
2
v
2
c
p  m0 v
Clicker Question 4:
Peggy passes Ryan at velocity v. Peggy and Ryan both
measure the time it takes the railroad car, from one end
to the other,
other to pass Ryan.
Ryan The time interval Peggy
measures is ____ the time interval Ryan measures.
A. longer than
B. at the same as
C. shorter than
Clicker Question 4:
Peggy passes Ryan at velocity v. Peggy and Ryan both
measure the length of the railroad car, from one end to
the other
other. The length Peggy measures is ____ the length
Ryan measures.
A. longer than
B. at the same as
C. shorter than