Problems on Orbits
Problem 1.1:
In
Solving the Orbits we derived the
equation:
From this, derive the expression we stated for
1/r. Hint: Define
y = 1/r
and use the fact that
= - .
[Solution]
Problem 1.2:
Using the expression we derived for
(1/r), show that this reduces to
x2 = y2 = k2 -2kεx + ε2x2, where
k = ,
ε = , and
cosθ = x/r.
[Solution]
Problem 1.3:
For
0 < ε < 1, use the above equation to derive the equation for an
elliptical orbit. What are the semi-major and semi-minor axis lengths? Where
are the foci?
[Solution]
Problem 1.4:
What is the energy difference between a circular earth orbit of radius
7.0×103 kilometers and an elliptical earth orbit with apogee
5.8×103 kilometers and perigee
4.8×103 kilometers. The mass
of the satellite in question is 3500 kilograms and the mass of the earth is
5.98×1024 kilograms.
[Solution]
Problem 1.5:
If a comet of mass
6.0×1022 kilograms has a hyperbolic orbit around
the sun of eccentricity
ε = 1.5, what is its closest distance
of approach to the sun in terms of its angular momentum (the mass of the sun is
1.99×1030
kilograms)?
[Solution]
= - 
.
[Solution]
,
ε = 
, and cosθ = x/r.
[Solution]
