HOMEWORK SET 2, PHYS. 131
Solutions originally by Matt Pillsbury
Instructor:
Anthony Zee
Email:
[email protected]
TA:
Kevin Moore
Email:
[email protected]
5.4.
Work out the components of the fourvector
a
≡
d
u
/dτ
in terms of the
threevelocity
±
V
and the threeacceleration
±a
=
d
±
V /dt
to obtain expressions
analogous to (5.28). Using this expression and (5.28), verify explicitly that
a
·
u
= 0.
First, note that
dγ
dt
=
1
(1

±
V
2
)
3
±
V
·
d
±
V
dt
=
γ
3
±
V
·
With (5.28) on p. 84 this gives
a
=
dt
dτ
d
dt
(
γ, γ
±
V
)
=
γ
2
±
γ
2
(
·
±
V
)
, γ
2
(
·
±
V
)
±
V
+
²
and
a
·
u
=

γ
5
(
±
V
·
)+
γ
3
(
±
A
·
±
V
γ
5
(
±
A
·
±
V
)
±
V
2
=
γ
5
(
·
±
V
)
³

1+
±
V
2
´
+
γ
3
(
·
±
V
)
=

γ
3
(
±
A
·
±
V
γ
3
(
±
A
·
±
V
)=0
1
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View Full Document5.7.
A particle is moving along the
x
axis. It is uniformly accelerated in the
sense that the acceleration measured in its instantaneous rest frame is always
g
, a constant. Find
x
and
t
as functions of proper time
τ
assuming that the
particle passes through
x
0
at time
t
= 0 with zero velocity. Draw the world line
of the particle on a spacetime diagram.
In the particle’s instantaneous rest frame, the last problem gives
a
t
±
=0
,a
x
±
=
g
Boost to the observer frame (where the particle has velocity
v
), using (5.9) from
p. 80 and (5.28) to get
a
t
=
gγv
=
gu
x
x
=
gγ
=
t
which is equivalent to
d
dτ
±
u
t
u
x
²
=
±
0
g
g
0
²±
u
t
u
x
²
This system of
ode
s is solved by
u
t
(
τ
)=
A
cosh(
gτ
)+
B
sinh(
)
,u
x
(
A
sinh(
B
sinh(
)
The particle is at rest at
τ
= 0, so
A
= 1 and
B
= 0. Then we have
t
(
τ
³
dτ u
t
=
g

1
sinh(
)
x
(
τ
³
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 Spring '08
 KEVIN
 Physics, Acceleration, Work, General Relativity, Special Relativity, instantaneous rest frame

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