PH 301
Dr. Cecilia Vogel
Lecture 6
Review
Lorenz transformation
velocity transformation
Outline
relativistic momentum & energy
mass energy
Relativistic Momentum
p
mv
v
1  
c
2
m (sometimes mo) is rest mass
measured when object is at rest
v is object’s velocity
p is the object’s momentum.
This quantity is conserved in all collisions,
reactions, etc
F=ma?
What is mass?
Mass tells how hard object is to accelerate.
F = ma is a classical equation
Generally F = dp/dt
is equivalent to F=ma,
if p=mv, with m constant, and a = dv/dt
Classically dp/dt=m dv/dt – requires the
same force, for the same rate of change of v
 Not at high speed!
F ≠ ma
p
mv
v
1  
c
2
  mv
As speed changes, numerator and
denominator both change
dp/dt ≠ m dv/dt
As v gets close to c, object gets harder
and harder to accelerate
What is mass?
Rest mass (m) is mass measured
when at rest
 property of the object
 what you look up in text book
Relativistic mass (mrel) shows
how hard it is to accelerate the
object
increases with speed
p = mrelv
if use relativistic mass
mrel 
m
v
1  
c
2
Energy
Momentum increases with
speed, so does energy.
E = mrelc2
E
mc
2
v
1  
c
When v= 0
Energy is not zero
rest energy = mc2
2
  mc
2
Kinetic energy
is zero when v=0
K= mc2 - mc2
Momentum and Energy Units
Energy SI units:
 J = kgm2/s2 = CV
Momentum SI units:
 kgm/s
Mass SI unit
kg
Energy can also be given in eV
Momentum can also be given in eV/c
Mass can also be given in eV/c2
Momentum Change p 
mv
v
1  
To increase the speed of electron
c
from 0.1c to 0.2 c:
from 0.8c to 0.9 c:
MeV )(0.8c)
(0.511MeV c 2 )(0.1c)
(0.511
c2
p1 
p3 
2
1  0.1
1  0.82
p3  0.681 MeV/c
p1  0.05135MeV/c
p 4  1.055 MeV/c
p 2  0.1043MeV/c
p  0.374 MeV/c
p  0.053MeV/c
Same change in speed, but seven times as
much force needed.
2
Energy Change
E
mc 2
v
1  
c
To increase the speed of electron
from 0.8c to 0.9 c:
from 0.1c to 0.2 c:
(0.511 MeV c2 )(c 2 )
E1 
2
E3  0.8517 MeV
1   0.1
E1  0.5136 MeV
(K1  0.0026 MeV)
E 2  0.5215 MeV
E 4  1.1723MeV
E  0.32MeV
E  0.008MeV
Same change in speed, but forty times as
much energy needed!
2
Graphically
to ∞
It would require infinite force and
infinite energy to accelerate anything
with mass to the speed of light!
to ∞
Inertial Reference Frames
recall
special relativity (ch 2)
only true for
inertial ref frame is one in which
Newton’s Law of Inertia holds
not accel
no gravity (or “weak” like Earth’s)
To describe grav or accel
use General Relativity
Principle
 Recall the first Postulates of Special Rel:
 All laws of physics the same in all
inertial frames
const vel frame indistinguishable
from rest frame
The basic postulate of gen rel
The Principle of Equivalence
uniform gravity frame indistinguishable
from constant accel frame
equiv accel and g are equal in size, but
opposite dir
Equivalence
 Artificial Gravity
rotating spaceship, with centrip accel = g
feels like home, earth’s grav
Virtual Reality
tilted chair has grav down and back
feels like grav down, and accel forward
Car accelerating forward
you pushed down and back in chair
cup falls down and back
fuzzy dice hang down and back
He balloon floats up and forward
all as if grav had a backward component
Effect on Light
Effect of grav on light identical to effect of
accel on light.
Light traveling perpendicular to grav field:
consider accel frame
you move a small dist rel to light at
first, then larger and larger distances
in your frame, light moves small
distance, then larger and larger
light follows curved path
see handout
Effect on Light
Effect of grav on light identical to effect of
accel on light.
Light traveling parallel to grav field:
consider accel frame
the frame is moving slowly when
light emitted
moving faster when received
as if sender and receiver are moving
relative to each other
Doppler shift
see handout
Gravity
Light’s path is bent by gravity,
light from a star behind the sun travels
a curved path, so appears at a different
place
gravitational lensing
Light retardation,
pulse sent to Venus
is slowed as it passes the Sun
all of time is slowed by grav not just the
freq of light
Black Hole
very large stars collapse
their gravity is stronger than the Pauli
repulsion of neutrons
the star continues to collapse
no known force can stop it
collapses to zero volume (?)
infinite density (?)
PAL – Classical and Relativistic
Momentum
 In classical mechanics, momentum is
proportional to velocity, so if you double the
velocity, you double the momentum. Let’s see
how this goes at high speed:
1 a) By what factor does the momentum
change when an object’s speed is doubled
from 0.05 c to 0.1 c?
1 b) By what factor does the momentum
change when an object’s speed is doubled
from 0.3 c to 0.6 c?
1 c) The faster you go, the (worse or better)
classical physics fits reality.
PAL – Classical and Relativistic
Kinetic Energy
 In classical mechanics, kinetic energy is
½mv2. Let’s see how this goes at high speed:
2 a) What is the percent error you get by using
the classical equation instead of the relativistic
equation for an object moving at 0.05 c?
2 b) What is the percent error you get by using
the classical equation instead of the relativistic
equation for an object moving at 0.6 c?
2 c) The faster you go, the (worse or better)
classical physics fits reality.
Spacebarn Paradox
The spaceship is 40 m long in its own
frame. The spacebarn is traveling at
about 0.866c relative to the ship, so it is
only 10.01 m long in the ship’s frame.
In the ship’s frame, the ship does NOT fit
in the spacebarn. In the ship’s frame,
will the ship get hit by the doors?