# Gravitation: Orbits

### Problems on Orbits

Problem : 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 = - .

Problem : Using the expression we derived for (1/r) , show that this reduces to x 2 = y 2 = k 2 -2kεx + ε 2 x 2 , where k = , ε = , and cosθ = x/r .

Problem : 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?

Problem : 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.

Problem : 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)?