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Problem  Tangential Impulse Orbit Change
(a)
From the result of GPS Problem 211 (2.9 in 2
nd
ed.) we can relate the
change in angular momentum incurred by the cometstar system to the strength
of the tangential impulse
S
,
(
μr
2
·
θ
)
f
−
(
r
2
·
θ
)
i
=
l
−
l
0
=
S,
(1)
where the subscripts
i
and
f
refer to values just before and after the impulse,
respectively. Since the impulse occurs at perihelion,
r
=
r
p
,where
·
r
=0
,wecan
use the energy equation for the parabolic orbit to relate
r
p
and
l
0
.
The general
expression for the energy of the orbit is
E
=
μ
·
r
2
2
−
k
r
+
l
2
2
μr
2
.
(2)
For the parabolic orbit,
E
=0
, and we have
0=
−
k
r
p
+
l
2
0
2
μr
2
p
,
(3)
which yields
r
p
=
l
2
0
2
μk
.
(4)
The new circular orbit has radius
r
p
, and we can use the circular orbit condition
to relate
r
p
and
l
,
f
ef
(
r
p
)=0=
−
k
r
2
p
+
l
2
μr
3
p
,
(5)
which yields,
r
p
=
l
2
μk
.
(6)
Comparing Eqs.(4) and (6), we
f
nd
l
2
=
l
2
0
2
,
(7)
implying two possible values for
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This note was uploaded on 12/19/2010 for the course PHYSICS ph 409 taught by Professor Wilemski during the Fall '10 term at Missouri S&T.
 Fall '10
 Wilemski
 mechanics, Angular Momentum, Momentum

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