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Unformatted text preview: Physics
601
Homework
5­­­Due
Friday
October
8
 € € € 
 1. A
standard
result
in
undergraduate
relativity
is
the
velocity
addition
formula:
 v +v v = 1 2 .

This
typically
obtained
by
taking
the
product
of
two
Lorentz
 1 + v1v 2 transformations
and
the
doing
some
algebra.

A
more
straightforward
way
to
 γ γ vx 1 µ obtain
this
use
the
four
velocity:
 u = γ = .



Suppose
I
start
with
a
 γ v y 1− v 2 γ vz γ1 µ γ1 v1x 
.

 particle
moving
with
velocity
 v1
with
an
associated
four‐velocity
 u1 = γ1 v1y € γ1 v1z Suppose
one
boosts
to
a
new
frame
by
running
to
the
left

(‐x
direction)
with
a
 € γ 2 v 2 γ 2 0 0 €Λµ ν = v 2 γ 2 γ 2 0 0 

.

 velocity
which
corresponds
to
a
Lorentz
transformation
 0 0 1 0 0 0 1 0 ν The
four‐velocity
in
the
new
frame
is uν = Λµ ν u1 .
 a. From
the
transformed
four
velocity
find
 v x , v y , v z 
in
the
new
frame.
 v +v € b. For
the
case
where
 v1
is
entirely
along
the
x
direction,
show
that
 v = 1 2 .
 1 + v1v 2 € 
 € 
 € 2. Consider
relativistic
transformations
restricted
to
one
spatial
direction.

In
that
 € case,
the
velocity
can
be
specified
by
a
single
number
from
‐1
to
1
(in
units
with
 c=1).

It
is
convenient
to
introduce
the
“rapidity”
η 
with
the
property
that
 v = tanh(η) .

Note
that
while
v
is
restricted
from
‐1
to
1,
η goes
from
‐∞
to
∞.


 cosh(η) µ sinh(η) .
 a. Show
that
the
4‐velocity
is
given
by
 u = 0 0 b. Show
that
while
velocities
do
not
add
linearly
in
relativity
rapidities
do.
That
 is
one
obtains
the
relativistic
velocity
addition
formula
by
taking
 η = η1 + η2 .

 In
a
certain
sense
this
turns
out
to
be
just
the
hyperbolic
trig
version
of
the
 € angle
addition
formula
for
two‐dimensional
rotations.
 
 € 3. Relativistic
Kinematics:

Consider
the
elastic
scattering
of
two
particles
with
masses
 m1 
and
 m2 .

Suppose
initially
particle
1
is
at
rest
and
and
particle
2
approaches
it
 € 
 
 
 with
a
velocity
v
in
the
positive

z
directions
after
the
scattering
particle
1
goes
off
 making
an
angle
θ1 with
respect
to
the
z
axis.

The
purpose
of
this
problem
is
to
find
 expressions
for
the
magnitude
of
the
velocity
of
the
two
particles
and
the
angle
 made
by
the
second
particles
in
terms
of
v
and

θ1.

Do
this
by
first
going
to
the
 center
of
mass
frame
(with
zero
total
momentum)
where
energy
and
momentum
 conservation
tell
us
the
scattering
must
be
back
to
back
and
then
boosting
back
to
 the
lab
frame.


 
 4. A
rocket
fires
a
constant
rate
so
that
rocket’s
own
rest
frame
it
experiences
an
 acceleration
in
the
+x
direction
with
a
magnitude
of
g
(i.e.
an
observer
in
the
rocket
 ship
would
feel
an
artificial
gravity
pushing
them
in
the
negative
x
direction
with
a
 force
of
mg).
 a. Show
that
the
equation
of
motion
for
the
rocket
is
given
by
 0 − g 0 0 du µ µν µν g 0 0 0 
.

To
do
this
you
need
to
show
 = G uν with G = 0 0 0 0 dτ 0 0 0 that
the
acceleration
in
the
commoving
frame
is
correct
and
that
the
form
 d ( uµ u µ ) correctly
gives
 = 0 
as
required
from
the
definition
of
 u µ .
 dτ € b. Work
in
a
frame
for
which
the
rocketship
starts
at
rest
at
the
orgin
at
t=0.


 Find
an
expression
for u µ (τ ) .
 € c. Use
the
result
of
b.
to
compute
 x µ (τ ) .
 € d. Find
x(t).
 € € 
 ...
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This note was uploaded on 12/29/2011 for the course PHYSICS 601 taught by Professor Hassam during the Fall '11 term at Maryland.

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