A satellite of mass 2500 kg is orbiting the Earth i n an elliptical orbit. At the farthest point from the Earth, its altitude is 3600 km, while at t he nearest point, it is 1100 km. Calculate the...
Let's start with the equations you will need for this problem.
Gravitational potential energy, U:
where: G = gravitational constant, m = satellite's mass, and M = Earth's mass
Aphelion is where the satellite is furthest away:
`r_1= R_E + 3600 km=9.967*10^6 m`
Perihelion is where the satellite is the closest:
`r_2= R_E + 1100 km=7.467*10^6 m`
Conservation of Energy:
`K_1+U_1 = K_2+U_2`
Conservation of Angular Momentum:
`mv_1 r_1=mv_2 r_2`
Solve system of equations for v2.
`v_2^2 = 2GM*(1/r_1-1/r_2)/((r_2/r_1)^2-1)`
Plug in numerical values to and solve to get the velocities.
`v_1 = 4387.3 m/s` at aphelion, and `5856.2 m/s` at perihelion.
Next, plug in either position 1 or position 2 values to get E at BOTH aphelion and perihelion of `E=K+U=1/2*m*v_1^2-G*m*M/r_1=7.603*10^10 J`
Your angular momentum will also be the same for both aphelion and perihelion
Therefore this problem was completly solved by the conservation of energy and angular momentum.
I included a link that attempts to solve this same problem in more detail. There numerical values are incorrect because they are missing a factor of two when they solve for the velocity.