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1.
Setup: 1 open four browser
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for class:
http://www.physics.byu.edu/Courses/CourseLis
t.aspx
2. For twin
http://physics.syr.edu/courses/modules/LIG
HTCONE/LightClock/#twins
3. MU 43:14 (velocities) 11.22 minute
1. MU 43:19 Twin
4. MU 44: 9’15” to 14’ 30” billiards
5. Looking at appearances:
http://www.anu.edu.au/Physics/Savage/TEE/site/
Twin paradox, Relative
Velocity, and Momentum
Class 3: Everything should be made
as simple as possible but not simpler.
A. Einstein.
Which of these are invariant
quantities? time, distance, velocity, c,
frequency, charge, mass, momentum,
force?
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class.)
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4. Today unless noted otherwise questions
are participation points (pp).
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Did you complete at least 70% of Chapter 1: 5-7?
Not pp
A. Yes
B. No
Einstein once gave a talk on some new idea of his
at a meeting of the German Physical Society. After
he had spoken and the chairman had respectfully
invited questions a young man got up at the back of
the hall and said something like ‘What Professor
Einstein has told us is not so stupid.
But the second equation does not strictly follow from the first.
It needs an assumption, which is not proved, and moreover it
is not invariant the way it should be. . .’ Everyone in the hall
had turned round and was staring at the bold young man.
Except Einstein; he was facing the blackboard and thinking.
After a minute he turned round and said ‘What the young man
in the back has said is perfectly correct; you can forget
everything I have told you today!’
Plan: Reading was 1.5-1.7
1. Review: EXAMPLE 1.11 (page 44 of
book) or Velocities and space-time
diagrams 43:14 starts at min 11.22'
2. Twin paradox MU or as viewed in light
cone space time diagram.
http://physics.syr.edu/courses/module
s/LIGHTCONE/LightClock/#twins
3. E & M effects: Story & Problem
4. “Relativistic momentum changes” and
discussion of mass as invariant.
Relative Velocity
u v
ux 
'
u xv
1 2
c
'
x
Shall we do velocity examples?
A. Yes, do one or more
B. Yes, let it be one from HW.
C. No, but let’s discuss formulae etc.
D. I want to see MU 43 11.22’ on this.
E. No, let’s go on.
Some of the issues to discuss are how do
we get the signs right? How do we
read the notation? How looks in ST?
4. (4 pts) The nucleus of a particular atom, initially
at rest in the laboratory system, is unstable and
disintegrates into two particles. Particle one
moves to the left at a speed of 0.80c, and
particle two moves to the right at a speed of
0.98c.
(a) What is the velocity of particle one with respect
to an observer at rest with particle two?
(b) An observer at rest with respect to the
laboratory system finds that both particles are
unstable. Particle one decays after 6.6 μs.
Particle two decays after 6.0 μs. What are the
lifetimes of the two particles from a reference
frame in which particle two is at rest?
Relativistic Addition of Velocities
Galilean relative velocities cannot be applied
to objects moving near the speed of light.
But 1st step is Galilean. vab = vad + vdb
• Einstein’s modification is
v ab
v ad  vdb

v advdb
1
c2
– The denominator is a correction based
on length contraction and time dilation
– Vab is velocity of a in ref. frame of b, etc.
1. In classical physics, what
is the velocity of one of the
objects in the reference frame
of the other object? (pp)
0.5 c→ ←0.99 c
A. 0.49c
B. 1.49c
Why can’t it be 1.49 c really?
• What is relativistic velocity?
• Vab= [0.5c+0.99c]/
[1+(.5)(.99)/c2]= 0.9966c
• Another example?
2. In classical physics, what
is the velocity of one of the
objects in the reference frame
of the other object? (pp)
a 0.99 c→ b 0.98 c→
A. 0.01c
B. 1.97c
What is velocity of a in b, Vab?
a 0.99 c→ b 0.98 c→
v ad  vdb

v advdb
1
2
c
• We are “d” vab
• Vad = 0.99,
• Vbd =0.98c;
• But Vdb= -Vbd = -0.98c
• Vab=[0.99c -0.98c]/[1+(0.99)(-0.98)/c2]
= 0.34c
3. See ex. 1.11 in text
Spacecraft Alpha α is moving with v=0.90c with respect to the
earth. If spacecraft Beta β passes α at relative speed 0.50c in
the same direction, what is speed of β with respect to earth?
I am going to have them both going in the +x direction.
_____________ β→ _____ α→
vab = (vad + vdb)/(1+vadvdb /c2); here a is β and b is earth. d is α.
Now go in and put in names for symbols with goal of getting the
signs right.
vad is speed of β with respect to α= 0.50c. vdb is speed of α with
respect to earth = 0.90c.
βab = (0.5+0.9)/(1+0.5*0.9) =1.4/(1.45) = 0.97; I use βab ≡ vab/c
Many people use the expression “Laboratory Frame” where
“earth” is used in this problem. (1.11 in book)
How shall we review twin paradox?
Facts:
• Both see that the others clock
(biology) is running slow on way out.
• The fact that Henry turns around
makes his journey asymmetric.
Quiz: A. diagrams, B. See Mechanical
universe 43:19; C. No MU. I saw it but
want to see more web items.
D. Several of above E. go on.
Fig 39-10, p.1257
Fig 39-12, p.1259
Here is an electronic rendering of
“twin-paradox”. It is
like example 1.4 in text (page 19)
1. http://physics.syr.edu/courses/modul
es/LIGHTCONE/LightClock/#twins
Shall we review the Barn-Pole Paradox?
(Simultaneity) A. Yes B. No
The observer (Albert) in the “barn stationary
frame” sees an instant when the shortened
pole is all the way in the barn.
The runner (Henri) with the pole sees the
shortened barn headed at him. There is
no instant when the pole is all the way in
the barn.
Why no splinters? He claims that there is a
period of time when the leading edge of
the pole is out the back door and the
trailing edge still is not in the front door of
the barn. WHO IS RIGHT?
Review of barn paradox
• Albert sees that there is a way to
have both doors closed for an
instant.
• Henri sees that the barn is length
contracted, not his pole, and thus
there is no way to have both
doors closed without the pole
crashing into one or other.
What Albert
photographs
(using an
imagined
kind of
instantaneou
s light). Barn
doors are
the green.
Pole ends
are marked
with cyan.
Fig 39-13a, p.1261
They switched colors. Naughty
What
runner
“sees”.
His pole
ends are
marked
with the
green.
(Vertical
in time.)
Barn
doors are
in cyan.
Fig 39-13b, p.1261
Resolution? Events are at different
times for Henri than Albert.
• The observer (Albert) in the “barn frame” sees
an instant when the shortened pole is all the way
in the barn.
• The runner (Henri) with the pole sees the
shortened barn headed at him. There is no
instant when the pole is all the way in the barn.
Why no splinters? He claims that there is a
period of time when the leading edge of the pole
is out the back door and the trailing edge still is
not in the front door of the barn.
Both are RIGHT: Simultaneity depends on
reference frame
Section 1.6 (Why was E&M
Einstein’s way to get to relativity?)
Did you get the email about viewing MU?
MU 43: min. 1’ 43”- 5’40” and 24’ 15”
Please do this Quiz:
I viewed MU 43: 1’ 43”- 5’40”
A. Yes. B. No, but I will.
Quiz counts
E&M: A natural way for relativity
Consider the case of cars headed north to SLC at
Each has a charge of 50 nC. 1 car passes me
standing on the side of the rode each second.
What kind of field could be measured?
A. An Electric B. A Magnetic C. Both D. Neither
Now suppose you are in one of the cars.
What kind of field could be measured riding in car?
Same choices.
More ideas at
http://www.cavehill.uwi.edu/fpas/cmp/online/p2
0b/P20B%20Notes.htm pp. 65-67.
Section 1.7: Force & momentum
• F=ma? = mdv/dt?
• The trouble is that v depends on
frame and time does too!
• Key idea: F=ma when v=0; but is
there another way to think about
force?
• Rewrite F= dp/dt. (This is really what
Newton had in the first place.)
• p=γmv
Watch Out for “Relativistic Mass”
• Some older treatments of relativity maintained
the conservation of momentum principle at high
speeds by using a model in which the mass of
the particle increases with speed. You might
still encounter this notion of “relativistic mass” in
your outside reading, especially in older
books. Be aware that this notion is no longer
widely accepted and mass is considered as
invariant, independent of speed. The mass of
an object in all frames is considered to be the
mass as measured by an observer at rest with
respect to the object.
Please do this Quiz: I viewed the
MU online on relativistic billiards
(44: 9’15” to 14’ 30”)
A.Yes.
B. No, but I will.
C. No, and I want us to
view it now instead of
rest of material.
Relativistic Linear Momentum
To account for conservation of momentum in
all inertial frames, the definition must be
modified to satisfy these conditions
– The linear momentum of an isolated particle
must be conserved in all collisions
– The relativistic value calculated for the linear
momentum p of a particle must approach the
classical value mu as u approaches zero
mu
p
 γmu
u2
1 2
c
– u is the velocity of the particle, m is its mass
Relativistic Form of Newton’s Laws
• The relativistic force acting on a particle
whose linear momentum is p is defined as
F = dp/dt.
• This preserves classical mechanics in the
limit of low velocities.
• It is consistent with conservation of linear
momentum for an isolated system both
relativistically and classically.
• Looking at acceleration it is seen to be
impossible to accelerate a particle from
rest to a speed v  c. more below.
If we apply a constant force on
an object, the acceleration of the
object, as it approaches the
speed of light, will _______. (pp)
A. decrease
B. increase
C.remain constant
Constant Force (begin at rest)
F=p/t ► p= Ft ► γmv =Ft
v<<c ► v increases
v→c ► γ increases
t= γmv/F
as v→c, ► γ →∞, ► t→∞
v never reaches c.
Speed of Light, Notes
• The “speed of light” is the
speed limit of the
universe. (Or is it the
other way around?)
• It is the maximum speed
possible for energy and
information transfer
• Any object with mass
must move at a lower
speed
• At some point look at appearances:
http://www.anu.edu.au/Physics/Savage/TE
E/site/
Next time: Relativistic Energy
• The definition of kinetic energy requires
modification in relativistic mechanics
• E = mc2 – mc2
– This matches the classical kinetic energy
equation when u << c
– The term mc2 is called the rest energy of the
object and is independent of its speed
– The term mc2 is the total energy, E, of the
object and depends on its speed and its rest
energy
Relativistic Energy –
Consequences
• A particle has energy by virtue of its mass
alone
– A stationary particle with zero kinetic energy
has an energy proportional to its inertial mass
– This is shown by E = K + mc2
• A small mass corresponds to an enormous
amount of energy
Energy and Relativistic
Momentum
It is useful to have an expression relating total
energy, E, to the relativistic momentum, p
– E2 = p2c2 + (mc2)2
When the particle is at rest, p = 0 and E = mc2
Massless particles (m = 0) have E = pc
– The mass m of a particle is independent of its
motion and so is the same value in all reference
frames
• m is often called the invariant mass
Lack of simultaneity is best
seen with long distances.
• Imagine signal being sent off from a
relativistic spacecraft part of way
between stars. (LOOK at figure on
overhead.)
• If ½ way in earth-star reference frame
received at same time.
• but c is c, so from spacecraft received
by star first since it is coming at you.
Simultaneous Events
• MU 42:20