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!"#$%&'(()*(+#(.(*&!&($',-$)"!(13(@./+/%."(&%/.6>+*2(*1(.//(
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%:%-*#(13(#+4A/*."%+*28(64%(9+/.61"(."9(#,.-%(
-1"*&.-61"?(
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D24=30%-E)%4-)&$-7)F22:)D24)(02?)
=26/<F)G6HH).I)D4%=-(8)
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=22<)G(2)&$%&)723).230:)1-)723<F-4)&$%<)7234)142&$-4)%Y-4)&$-)&4/,I)
"t 0 = "t 1- v 2 /c 2 = "t 1-12 2 /(3 #10 5 ) 2 = "t 1-12 2 /32 #1010
Sorry, you will look the same still..
!
!
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!
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P"))KD)2<-)D4%=-)=26-()/<)%).-4&%/<):/4-.52<E)&$-)2&$-4)D4%=-)
=3(&)1-)=26/<F)/<)&$-)2,,2(/&-):/4-.52<)G/&)/()%00)4-0%56-I"))
B$-<)$2?):2)K):-&-4=/<-)?$/.$)2<-)($230:)1-)&$-)!&Q)%<:)
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\%2(-1"#*."*'((>%/1-+*2(13(*0%(#0+,(
v=
!
L0
L
=
"t "t 0
]2<&4%.&-:)W-<F&$)
L = L0
"t 0
v2
= L0 1# 2
"t
c
!
"t 0
v2
L = L0
= L0 1# 2
"t
c
!<.+"8(+3(>(__(-8(*0%(/%"<*0(-1"*&.-61"(+#(-/1#%(*1(U?((G0.*(+#($02(
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!
DA,,1#%(>(T-8(*0%(a"./(-1"*&.-*%9(/%"<*0(+#(U(4%*%&#?(
Doc is right, Barney’s car that moves at the speed of
light will disappear!!
^&$-4)-;-.&(S))+,%.-)]2<&4%.52<)2<07)?24>()%02<F)&$-)0/<-)
2D),42,%F%52<"))+3,,2(-)?-)4-,42:3.-)&$/()/=%F-)/<)%)D%(&)
=26/<F)(,%.-($/,"))B$-)4-(30&)?/00)1-)U?%4,-:V)/<)_)
:/4-.52<)2<07"))`3<<7)-;-.&()$%,,-<)G-"F"E))Ra@Wb+)2D)
Image observed by a person on Earth!
/<.0/<-)$%6-)/<.4-%(-:LI)
>(
On Earth!
Suppose the same tree and people
are placed inside a spaceship
E%/.6>+#6-(!99+61"(13(]%/1-+6%#(
Z(
Q(
@./+/%."(E%/.6>+*2'(((>QXT(>QZ(b(>ZX(
B+"#*%+"(D,%-+./(E%/.6>+*2'(
v 23 =
v 21 + v13
v v
1+ 21 213
c
)3(*0%(&1-H%*(+#(.-*A.//2(.(P%.4(13(/+<0*(8($0.*(+#(+*#(#,%%9(
&%/.6>%(*1(1Pc%-*(X(5.#*%&1+98(A"41>+"<7(
v 23 =
c + v13 c(1+ v13 /c)
=
=c
cv13
1+
v
/c
13
1+! 2
c
This provides theoretical basis for the uniform speed of light in vacuum.
X(
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E%/.6>+*27(
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414%"*A4(#A-0(*0.*(-1"#%&>.61"(13(414%"*A4($+//(P%(
%"#A&%9?(
(2) When two fast moving bodies (speed ~ light) collide,
they may gain more speed, but speed > c will not happen.
Hence, a different formula from Galilean momentum must
be used.
c-0%56/(5.)d2=-<&3=)
!
p=
!
mv
1" v 2 /c 2
)=2=-<&3=)
The classical and !
relativistic
momentums are essentially the same
at speeds << c. As speed increases,
the relativistic momentum increases
more rapidly than the classical
momentum
Example 1: At what speed does the
classical momentum, p=mv, give an
error when compared with relativistic
momentum, of (a) 1.0% and (b) 5.0%.
Relativistic
Momentum
Classical
Momentum
U?Q( U?f( U?S( U?g( Z?U(
((((>(h-(
Problem 2: A satellite initially at rest in deep space, explodes
into two pieces. One piece has a mass of 150 kg and moves
away from the explosion with a speed of 0.76c. The other
piece moves away from the explosion in the opposite direction
with a speed of 0.88c. Find the mass of the second piece of
the satellite.
Insights from Example 2:
Suppose we do classical calculation, we shall have
p = mv
0 = m1v1 " m 2v 2
0 = 150 " 0.76c # m 2 " 0.88c
150 " 0.76c
classical
mass m 2 =
= 129.50 kg
!
0.88c
!
relativistic mass = 95 kg
!
!
Classical results over-estimates the mass due to the
omission of factor 1" v 2 /c 2
! this relativistic equation of momentum
In fact,
is sometimes thought of in terms of masses
that increase with speed:
!
&
! #
m0
p =%
(v
$ 1" v 2 /c 2 '
!
m0=“rest mass”
!
p=
!
mv
1" v 2 /c 2
#
&
m0
m =%
(
$ 1" v 2 /c 2 '
!
!
Simple Consequences of Changing Mass:
(1)! It is impossible to accelerate to a speed v > c.
When v = c,
#
&
m0
m =%
(=)
2
2
$ 1" v /c '
F = ma = " # a = "
or a = F/m = 0
!
Infinite force and/or having
essentially 0 acceleration when
v~c, that means an object will
never creep over the speed of
c if it stops to accelerate.
Final Exam End Point
!
(2) Relativistic mass ----- > Relativistic total/kinetic energies
When work is done on an object, it will not only increase speed (i.e.,
increase kinetic energy), but also increase Mass.
Famous Einstein Equation:
when it moves with speed v,
For an object with rest mass m0,
#
& 2
m0
2
E =%
(c = mc
2
2
$ 1" v /c '
What does it mean?
Increasing energy ----- > increasing mass
This means that a compressed spring
gains mass since elastic potential
energy change E=0.5 kx2
If v=0, total energy IS NOT 0, as there is rest energy
E 0 = m0c 2
(in Joules)
Example: Rest energy of an 0.12 kg apple
E 0 = m0c 2 = 0.12 " (3.0 "10 8 ) 2 = 1.1"1016 J
This is enormous!!! If all of this mass is being converted
efficiently into a 100 W light bulb, the latter could stay lit for 10
!
million years (Fission --- > nuclear energy is based on this) !
!
Example: The Prodigal Sun
Energy is radiated by the sun at the rate of about 3.92 x 1026
W. Find the corresponding decrease in the Sun’s mass for
every second it radiates. Will it run out soon?
Solution: Energy is radiated continuously by the sun at the
rate of
Power = 3.92 "10 26 W
Loss of Energy in 1 sec = 3.92 "10 26 J
The corresponding decrease in mass, according to
!
the relation
E=mc2, is given by
"m = "E /c 2
! = 3.92 #10 26 /(3.0 #10 8 ) 2 = 4.36 #10 9 kg
"m
Insights: The Sun loses a large amount of mass
! going to be a problem (run
each second, but is that
out soon)?
30
!
Mass of the Sun : m = 1.9 "10 kg
Total 'burn' time : t =
!
!
=
1.9 "10 30 kg
= 4.36 "10 20 sec
9
4.36 "10 kg/s
4.36 "10 20 sec
= 1.38 "1013 year
3600sec/h " 24 h/day " 365 day/yr
Age of sun/solar system: 4.2 x 109 year
!
The baby Sun
Relativistic Kinetic Energy
Based on the velocity dependent mass and relativistic
momentum formula, one is temped to generalize as
follows:
mv 2
m0
WRONG: K_relative =
(where m =
)
2
1- v 2 /c 2
Correct derivation:
m0c 2
E total =
!
1" v 2 /c 2
Now, since total energy = rest energy + kinetic
m0c 2
E total =
= E rest + K
2
2
!
1" v /c
Rest energy is:
E rest = m0c 2
K=
!
!
m0c 2
2
1" v /c
2
" m0c 2
!
Matter & Antimatter
A Consequence of mass & energy equivalence is
the existence of antimatter
For every elementary particle (e.g., electron, proton, neutron)
known to exist, there is a corresponding antimatter particle that
has precisely the same mass but the exactly the opposite charge.
When colliding, they annihilate one another (no more charges).
Also, to conserve
Annihilation of matters
Positron
(NOT PROTON)
Electron
+
-
In this vanishing act, below are
conserved:
(1)! Momentum
(2)! Angular momentum
(3)! Total energy
Antimatter Generation + Usage
Cosmic rays
(a natural source,
High energy)
Cosmic rays from outer space were the first high-energy particles ever
studied. A few cosmic rays pass through everyone every second of every
day, It is difficult to work out the exact origin of cosmic rays, but many are
thrown into space by supernovae, the huge explosions of dying stars.
Cosmic rays hitting the outer atmosphere are mainly fast-moving, highenergy protons. As they propagate towards the Earth, they collide with
atoms in the air. Some of the collision energy reappears as the mass of
new pairs of particles and antiparticles. Cosmic rays are thus a natural
source of antiparticles.
Antimatter Generation + Usage
Use the emission of gamma rays to detect
tumors (very much the same way as using
seismic waves to detect gas pockets)
Low energy
Today antimatter is used every day in medicine for brain scans. When
electrons and positrons meet, at high energies, they can rematerialize as
new particles and antiparticles, but at low energies, the electron-positron
annihilations can used to reveal the workings of the brain in the technique
called Positron Emission Tomography (PET). In PET, the positrons come
from the decay of radioactive nuclei incorporated in a special fluid injected
into the patient. The positrons then come in contact with electrons in nearby
atoms. As the electrons and positrons are almost at rest when they come
together, there is not enough annihilation energy to make even the lightest
particle and antiparticle, so the energy emerges as two gamma rays that
shoot off in opposite directions and can then be detected and analyzed to
determine various aspects of the brain.