Physics 2D Lecture Slides
Lecture 4
Jan. 11, 2010
Lorentz Velocity Transformation Rule
S and S’ are measuring
ant’s speed u along x, y, z
axes
x2' ! x1'
dx '
In S' frame, u x' = '
= '
'
t2 ! t1
dt
v
dx = " (dx ! vdt), dt ' = " (dt ! 2 dx)
c
dx ! vdt
u x' =
, divide by dt
v
dt ! 2 dx
u
c
ux ! v
u x' =
vux
1! 2
c
For v << c, u x' = ux ! v
'
S
S’
v
(Galilean Trans. Restored)
Velocity Transformation Perpendicular to S-S’ motion
v
dy ' = dy, dt ' = ! (dt " 2 dx)
c
dy '
dy
'
uy =
=
v
dt '
! (dt " 2 dx)
c
divide by dt on RHS
u =
'
y
uy
v
! (1 " 2 ux )
c
There is a change in velocity in the
direction # to S-S' motion !
Similarly
Z component of
Ant' s velocity
transforms as
uz
'
uz =
v
! (1 " 2 ux )
c
Inverse Lorentz Velocity Transformation
Inverse Velocity Transform:
ux ' + v
ux =
vux '
1+ 2
c
'
As usual,
uy
replace
uy =
v '
v⇒-v
! (1 + 2 ux )
c
'
uz
uz =
v '
! (1 + 2 ux )
c
Does Lorentz Transform “work” For Topgun ?
y’
Two rockets A &B travel in
opposite directions
S’
-0.85c
y
S
B
0.75c
A
x’
O’
O (Earth guy)
x
An observer on earth (S)
measures speeds = 0.75c
And 0.85c for A & B
respectively
What does A measure as
B’s speed?
Place an imaginary S’ frame on Rocket A ⇒ v = 0.75c relative to Earth Observer S
Consistent with Special Theory of Relativity
Example of Inverse velocity Transform
Biker moves with speed = 0.8c
past stationary observer
Throws a ball forward with
speed = 0.7c
What does stationary
observer see as velocity
of ball ?
Speed of ball relative to
stationary observer
uX
?
Place S’ frame on biker
Biker sees ball speed
uX’ =0.7c
Hollywood Yarns Of Time Travel !
Terminator : Can you be seen to be born before your mother?
A frame of Ref where sequence of events is REVERSED ?!!
u
ff
fro
eo
ak
S
in
(,)(,)xtxt
22''22
F
'
1
ve
*
$ v !x ' !t ' = t " t = # , !t " & 2 ) /
% c (.
+
Reversing sequence of events 0 !t ' < 0
'
2
rri
(,)(,)xtxt
11''11
Ia
m
SD
S’
It
S
I Can’t be seen to arrive in SF before I take off from SD
u
S
S’
(,)(,)xtxt
11''11
22''22
(,)(,)xtxt
*
$ v !x ' !t ' = t " t = # , !t " & 2 ) /
% c (.
+
For what value of v can !t ' < 0
v !x
v !x v u
!t ' < 0 0 !t < 2
01 < 2 = 2
c
c !t
c
v c
0 >
0 v > c : Not allowed
c u
'
2
'
1
Watching an Inelastic Collision between two putty balls
Relativistic Momentum and Revised Newton’s Laws
Need to generalize the laws of Mechanics & Newton to conform topmu=!!
Lorentz Transform
and the Special theory of relativity: Example :
S
P = mV –mV = 0
P=0
Before
v
1
v
U=0
2
1
2
After
u1 ! V
u !V
!2V
U !V
= 0, u '2 = 2
=
,
U
'
=
= !V
2
u1V
u2V
UV
V
1
!
1! 2
1! 2
1+ 2
c2
c
c
c
!2mV '
'
'
'
pbefore = mu1 + mu2 =
, pafter = 2mU ' = !2mV
2
V
1+ 2
S’
c
p'before " p'after
u1' =
1
u1’=0
Before
u2 ’
2
U’
1
2
After
Definition (without proof) of Relativistic Momentum
p=
mu
1 ! u 2 / c2
With the new definition relativistic momentum is
conserved in all frames of references : Do the exercise
New Concept
Rest Mass is mass of object in frame in which it at rest.
! = 1 / 1 " u 2 / c2
u is speed of object, NOT reference frame.
Nature of Relativistic Momentum
m
u
!
p=
!
mu
1 ! (u / c)2
!
= " mu
With the new definition of
Relativistic momentum
Momentum is conserved in
all frames of references
Relativistic Force & Acceleration
!
p=
!
mu
1 ! (u / c)2
!
= " mu
Relativistic
Force
And
Acceleration
Reason why you cant
quite get up to the speed
of light no matter how
hard you try!
!
%
! dp! d "
mu
F=
= $
' use
dt dt $# 1 ! (u / c)2 '&
)
m
mu
F=+
+
+ 1 ! (u / c)2
2
1
!
(u
/
c)
+*
) 2
2
2 ,
mc
!
mu
+
mu
. du
F=+
3/ 2 .
+ 2
2
dt
c
1
!
(u
/
c)
.*+
(
(
)
m
F=+
+
2
+* 1 ! (u / c)
)
3/ 2
d du d
=
dt dt du
,
!1 !2u . du
( ( )( 2 )
2 c . dt
.-
)
,
. du : Relativistic Force
3/ 2 .
dt
.!
! du
Since Acceleration a =
,[rate of change of velocity]
dt
!
3/ 2
F
!
)1 ! (u / c)2 ,
/ a=
m*
!
Note: As u / c 0 1, a 0 0 !!!!
Its harder to accelerate when you get
closer to speed of light
(
)
A Linear Particle Accelerator
-
F
q
-E
d
+
Parallel Plates
E= V/d
F= eE
V
Charged particle q moves in straight line
!
!
in a uniform electric field E with speed u
! !
accelarates under force F=qE
!
!
3/ 2
3/ 2
!
2
2
du
F "
u %
qE "
u %
!
a =
=
1! 2 ' =
1! 2 '
$
$
dt
m#
m #
c &
c &
larger the potential difference V across
plates, larger the force on particle
Under force, work is done
on the particle, it gains
Kinetic energy
New Unit of Energy
1 eV = 1.6x10-19 Joules
1 MeV = 1.6x10-13 Joules
1 GeV = 1.6x10-10 Joules
Your Television (the CRT type) is a Small Particle Accelerator !
Linear Particle Accelerator : 50 GigaVolts Accelating Potential
!
3/ 2
! eE
2
a= "#1 ! (u / c) $%
m
PEP-II accelerator schematic and tunnel view
Relativistic Work Done & Change in Energy
x2
X2 , u=u
x1 , u=0
W =
!
x1
! !
F .dx =
x2
!
x1
!
dp !
.dx
dt
du
!
m
mu
dp
dt
p=
#
=
,
3/
2
dt
u2
$
u2 '
1" 2
1" 2 )
&
c
c (
%
substitute in W,
du
u
m
udt
dt
# W =!
(change in var x * u)
3/ 2
0 $
u2 '
&1 " 2 )
c (
%
Relativistic Work Done & Change in Energy
X2 , u=u
u
W=
x1 , u=0
("
0
mudu
u %
$1 ! 2 '
# c &
2
3/ 2
=
mc 2
1/ 2
" u %
$1 ! 2 '
# c &
2
! mc 2
= ) mc 2 ! mc 2
Work done is change in energy (KE in this case)
K = ) mc 2 ! mc 2 or Total Energy E= ) mc 2 = K + mc 2
But Professor… Why Can’t ANYTHING go faster than light ?
K=
mc 2
! mc (
2
1/ 2
( K + mc )
" u2 %
$1 ! 2 '
# c &
" u2 %
( $1 ! 2 ' = m2 c 4 "# K + mc 2 %&
# c &
( u = c 1! (
2
2
)
,
+
.
2
mc
+
.
=+
1/ 2 .
2
+ "1 ! u % .
+* $ c 2 ' .#
&
!2
K
K
!2
+
1)
(Parabolic
in
u
Vs
)
2
2
mc
mc
As K / 0 , u / c
1 2
Non-relativistic case: K = mu ( u =
2
2K
m
2
Relativistic Kinetic Energy Vs Velocity
A Digression: How to Handle Large/Small Numbers
• Example: consider very energetic particle with very large Energy E
E
mc 2 + K
K
! =
=
=
1
+
mc 2
mc 2
mc 2
1/ 2
u #
1&
=
1
!
Now calculate u from 
%
(
c $ "2'
• Lets Say γ =
• Try this on your el-cheapo calculator, you will get u/c =1, u=c due
to limited precision.
• In fact u ≅c but not exactly!, try to get this analytically
3x1011,
In Quizzes, you are
Expected to perform
Such simple
approximations
When Electron Goes Fast it Gets “Fat”
Total Energy E = ! mc = K + mc
2
2
New Concept
Rest Mass = particle mass when its at rest
Relativistic Kinetic Energy & Newtonian Physics
Relativistic KE K = ! mc 2 " mc 2
# Remember Binomial Theorem
&
%
(
nx
n(n-1)
% for x << 1; (1+x) n = (1+
+
x 2 +smaller terms) (
%$
('
1!
2!
"
1
2
# u2 &
1 u2
) When u << c, %1- 2 (
* 1+
+ ...smaller terms
2
2c
$ c '
1 u2
1 2
2
2
so K * mc [1 +
]
"
mc
=
mu (classical form recovered)
2
2c
2
Total Energy of a Particle E = ! mc 2 = KE + mc 2
For a particle at rest, u = 0 " Total Energy E= mc 2
E=mc2 ⇒ Sunshine Won’t Be Forever !
Q:
Solar Energy reaches earth at rate of 1.4kW per square
meter of surface perpendicular to the direction of the sun.
by how much does the mass of sun decrease per second
owing to energy loss? The mean radius of the Earth’s
orbit is 1.5 x 1011m.
r
• Surface area of a sphere of radius r is A = 4πr2
• Total Power radiated by Sun = power received by a sphere whose
radius is equal to earth’s orbit radius
E= mc2 ⇒ Sunshine Won’t Be Forever !
Total Power radiated by Sun
= power received by a
sphere with radius equal to
earth-sun orbit radius( r in figure)
r
sun
Plost
=
sun
Plost
P Earth
incident
Aearth! sun =
A
= 4.0 # 1026 W
P Earth
incident
A
2
3
2
11 2
4" rearth!
=
(1.4
#
10
W
/
m
)(4
"
)(1.5
#
10
)
sun
So Sun loses E = 4.0 # 1026 J of rest energy per second
E 4.0 # 1026 J
9
Its mass decreases by m = 2 =
=
4.4
#
10
kg per sec!!
8 2
c
(3.0 # 10 )
If the Sun's Mass = 2.0 # 1030 kg So how long with the Sun last ?
One day the sun will be gone and the solar system will not be a
hospitable place for life
E = ! mc 2 " E 2 = ! 2 m2 c 4
Relationship between P and E
p = ! mu " p 2 c 2 = ! 2 m2u 2 c 2
" E 2 # p 2 c 2 = ! 2 m2 c 4 # ! 2 m2u 2 c 2 = ! 2 m2 c 2 (c 2 # u 2 )
2 4
m2 c 2 2
m
c
2
2
2
2 4
=
(c
#
u
)
=
(c
#
u
)
=
m
c
2
2
2
u
c #u
1# 2
c
E 2 = p 2 c 2 + (mc 2 )2 ........important relation
For particles with zero rest mass like photon (EM waves)
E
E= pc or p =
(light has momentum!)
c
Relativistic Invariance : E 2 # p 2 c 2 = m2 c 4 : In all Ref Frames
Rest Mass is a "finger print" of the particle
Mass Can “Morph” into Energy & Vice Verca
• Unlike in Newtonian mechanics
• In relativistic physics : Mass and Energy are the same
thing
• New word/concept : MassEnergy , just like SpaceTime
• It is the mass-energy that is always conserved in every
reaction : Before & After a reaction has happened
• Like squeezing a balloon : Squeeze here, it grows
elsewhere
– If you “squeeze” mass, it becomes (kinetic) energy &
vice verca !
• CONVERSION FACTOR = C2
• This exchange rate never changes !