If a satellite is in a sufficiently low orbit, it will encounter air drag from t

Question

If a satellite is in a sufficiently low orbit, it will encounter air drag from the earth's atmosphere. Since air drag does negative work (the force of air drag is directed opposite the motion), the mechanical energy will decrease. If E decreases (becomes more negative), the radius r of the orbit will decrease. If air drag is relatively small, the satellite can be considered to be in a circular orbit of continually decreasing radius.

A satellite with mass 2360kg is initially in a circular orbit a distance 330km above the earth's surface. Due to air drag, the satellite's altitude decreases to 250km . Calculate the initial orbital speed. it is 7710

Calculate the increase in orbital speed. is 46

Then these are the parts that I need help with

Calculate the initial mechanical energy.

Calculate the change in kinetic energy.

Calculate the change in gravitational potential energy

Calculate the change in mechanical energy.

Calculate the work done by the force of air drag.

Explanation / Answer

It is generally so that the kinetic energy of a satellite orbiting with radius r is - 1/2 of the its potential energy, so the total energy is the negative of the kinetic energy.

1)
the velocity at a height of 330 km + 6371 km (=6701*10^3 m ) above the center of the earth is

v1 = ?(G*M/r1) = ?((6,674*10^-11)*(5974*10^24)/(6701*10^3)
v1 = 7713578.46 m/s

The kinetic energy = 1/2 mv^2 = (1/2)*2360*(7713578.46)^2

Ek1 = 7.02E16 J

= 7.02 x 10^16 J

The potential energy = - 2*Ekin = -m/2*G*M/r
Epot1 = - 1.40E1017 J

= -1.40  x 10^17 J

The total Energy = Ekin + Epot
Etot1 = - 7.02E16 J

= - 7.02 x 10^16 J
-------------------------------
2) same at height 250 + 6371 km = 6621*10^3 m:

v2 = 7760039.22 m/s

Ekin2 =7.11E16 J

= 7.11 x 10^16 J

Epot2 = - 1.42E17 J

= -1.42 x 10^17 J

Etot2 = - 7.11E16 J

= - 7.11 x 10^16 J
-------------------------------

Work done by air drag = Etot2 - Etot1

W = - 7.11E16 J + 7.02E16 J

W = - 0.09E16 J

= - 0.09 x 10^16 J

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