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In summary, the paradox of a satellite moving faster despite air friction is due to the conversion of gravitational potential energy to kinetic energy, resulting in a net increase in kinetic energy. This phenomenon can be observed when the satellite moves from a higher orbit to a lower one, and it can also spiral towards the Earth if the orbit is elliptical. However, this spiraling will not occur if the orbit is circular. The book also mentions that the assumption of a circular orbit is not valid in this scenario, as the initial velocity must not be tangential for the satellite to spiral towards the Earth without any energy loss.

## Homework Statement

A well-known paradox is that the air friction due to the outer layer of the Earth's atmosphere will reduce the total energy of the satellite and therefore the satellite will move slower. But it is observed that the satellite actually moves faster. Try to explain this phenomenon.

## The Attempt at a Solution

When the satellite is moving in a lower orbit, some gravitational potential energy of satellite is converted to KE of satellite, so KE of satellite increase and it moves faster when it travels to the Earth from space. Total energy decrease because some of the gPE of satellite is lost from satellite.

I've checked the answer. The answer said that gPE is converted to KE and KE gain of satellite is larger than KE loss due to air friction, so KE increases and it moves faster. In fact, is all gPE are converted to KE? Is there a part of gPE lost from the satellite?

Thank you very much!

If the satellite moves from a higher orbit to a lower one, some gravitational potential energy will be lost. For the satellite to loose all of it's potential energy, it would have to travel to the center of the Earth.

I think what the answer you have is saying is that the satellite has

PE1 = A
KE1 = B

It travels to a lower orbit and then has

PE2 = A - u
KE2 = B + u - w

and that

u > w

Where w is energy lost to friction.

Thank you.

The book also mentions that if the orbit is elliptical, the satellite can spiral towards the Earth even air friction is neglected. In fact why this happen? And why doesn't it happen if the orbit is circular? Thanks!

No, neglecting friction the orbit of a satallite is eliptical, NOT a spiral. Perhaps your book meant that a part of the orbit will pass through enough atmosphere that the air friction cannot be neglected. Please quote your book on this exactly.

A circle is a type of ellipse. Nothing special happens when the orbit is circular.

HallsofIvy said:
No, neglecting friction the orbit of a satallite is eliptical, NOT a spiral. Perhaps your book meant that a part of the orbit will pass through enough atmosphere that the air friction cannot be neglected. Please quote your book on this exactly.

A circle is a type of ellipse. Nothing special happens when the orbit is circular.

There is a question in the book:

A space shuttle of mass 10000kg coasts in a spiral path towards the Earth from P to Q with its engine shut off.
Take GMe=4.0*10^14 J/kg/m and radius of the Earth=6.4*10^6m and assum no frictional loss. Calculate the change in gravitational potential energy.

The solution for this question adds that:
We should not assume that the satellite is moving in a circular orbit. Otherwise, it would not spiral towards the Earth, unless there is frictional loss. Since there is no energy loss, we should assume that the initial velocity is not tangential.

Perhaps I get it wrongly Where does the book state that we have an orbit, never mind an elliptical one?

Hootenanny said:
Where does the book state that we have an orbit, never mind an elliptical one?

Actually there is a diagram next to the question. The diagram shows the Earth and two equipotential sufaces (only part of the surface is drawn, I can see two "curves" but I don't know whether it is circular or eliptical)
P and Q is two points on the two equipotential surfaces respectively.