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5 Gravitation
Fig. 5.20
For parabola
r =
2a
1 + cos θ
(2)
or cos θ =
2a
r
− 1
( 3 )
∴ tan
θ
2
=
1 − cos θ
1 + cos θ
=
R
a
− 1
( 4 )
where we have put r = R, the radius of earth’s orbit. Using (4) in (1)
t 0 =
2
3
2
G M
(2a + R)
√
R − a
(5)
t 0 is maximized by setting
dt 0
da
= 0. This gives
a =
R
2
(6)
Using (6) in (5) gives
t 0 (max) =
4
3
R 3
G M
=
2
3π
2π
R 3
G M
=
2
3π
T
where T = 2π
R 3
G M
= 1 year is the time period of the earth.
Thus t 0 (max) =
2
3π
years.
5 Gravitation
Fig. 5.20
For parabola
r =
2a
1 + cos θ
(2)
or cos θ =
2a
r
− 1
( 3 )
∴ tan
θ
2
=
1 − cos θ
1 + cos θ
=
R
a
− 1
( 4 )
where we have put r = R, the radius of earth’s orbit. Using (4) in (1)
t 0 =
2
3
2
G M
(2a + R)
√
R − a
(5)
t 0 is maximized by setting
dt 0
da
= 0. This gives
a =
R
2
(6)
Using (6) in (5) gives
t 0 (max) =
4
3
R 3
G M
=
2
3π
2π
R 3
G M
=
2
3π
T
where T = 2π
R 3
G M
= 1 year is the time period of the earth.
Thus t 0 (max) =
2
3π
years.