Satellite Orbiting: Speed, Period, Altitude Calc.

[squeak]

Homework Statement

Compute the speed and the period of a 240 kg satellite in an approximately circular orbit 610 km above the surface of Earth. The radius and mass of Earth are RE = 6400 km and ME = 6.0 × 1024 kg respectively. [5 marks]

Suppose the satellite loses mechanical energy at the average rate of 1.7 × 105 J per orbital revolution. Adopting the reasonable approximation that the satellite’s orbit becomes a circle of slowly diminishing radius, determine the satellites altitude, speed and period at the end of its 1500th revolution. What is the magnitude of the average retarding force on the satellite? [8 marks]

Homework Equations

U = -GMm/r

The Attempt at a Solution

I'm happy with how to do the first part of the question.
However the second part I'm struggling with.
I have ΔE = 1.7 x 10⁵ x 1500
I then think that the energy of the satellite initially = U = GMm/r+h where h is distance from Earth surface
energy final = GMm/r + h_f. This is where I have a problem as the worked solution I have been given states that E_initial = -½ GMm/r+h.
Can anyone explain why its multiplied by half?

 

[gneill]

Can anyone explain why its multiplied by half?

The total energy comprises both the kinetic and potential energy of the object in orbit.

Suppose you let $\mu = G M$, and let R be radius of the orbit. m is the mass of the orbiting object. Then the total specific mechanical energy is:

$ξ = \frac{v^2}{2} - \frac{μ}{r}$

Note: Multiply the specific mechanical energy by the mass m of the body to get the total mechanical energy in joules.

But the velocity of a body in circular orbit is given by $v = \sqrt{\frac{μ}{R}}$. Make the substitution for v in the specific mechanical energy formula.

 

[squeak]

The total energy comprises both the kinetic and potential energy of the object in orbit.

Suppose you let $\mu = G M$, and let R be radius of the orbit. m is the mass of the orbiting object. Then the total specific mechanical energy is:

$ξ = \frac{v^2}{2} - \frac{μ}{r}$

Note: Multiply the specific mechanical energy by the mass m of the body to get the total mechanical energy in joules.

But the velocity of a body in circular orbit is given by $v = \sqrt{\frac{μ}{R}}$. Make the substitution for v in the specific mechanical energy formula.

Thank you so much. That all makes sense now.