Energy Considerations When A Satellite Changes Orbit

[Bashyboy]

Homework Statement

A 1 034-kg satellite orbits the Earth at a constant altitude of 102-km.

(a) How much energy must be added to the system to move the satellite into a circular orbit with altitude 191 km?

(b) What is the change in the system's kinetic energy?

(c) What is the change in the system's potential energy?

Homework Equations

For (a): [itex]\Delta E = \frac{GM_Em}{2}(\frac{1}{r_i}-\frac{1}{r_f}[/itex]

The Attempt at a Solution

I have futilely attempted problem many vexing times; I am beginning to regard this as the bane of my existence.

For (a): [itex]r_i = R_E + 102000[/itex] and [itex]r_f = R_E + 191000[/itex], right?

[itex]\Delta E = \frac{GM_E \cdot 1034}{2}(\large \frac{1}{R_E + 102000}-\frac{1}{R_E + 191000}) = 890~MJ[/itex]



For (b) Supposedly there is some relation between gravitational energy and kinetic. I was just going to find the tangential velocity with respect to each orbit, and use those to find the change in kinetic energy. Although, seeing as this direct relationship will provide desired accuracy, what exactly is this relationship?

 

[gneill]

Your Relevant Equation deals with the kinetic energy of the system. For part (a) you'll need to account for the change in potential energy, too. What's the expression for gravitational potential energy?

You might want to take note that objects in higher circular orbits have lower orbital velocities; So check the order in which you specify the radius reciprocals in your KE formula.

Hint: An expression for the Total Specific Mechanical Energy of an orbit is:
$$\xi = \frac{v^2}{2} - \frac{\mu_E}{r}$$
where ##\mu_E = G M_E##

It combines the Kinetic and Potential energy contributions. The result is in J/kg, the energy per kg for the object in orbit. Multiply by the mass of the object in orbit to obtain the total energy in Joules.