Solving the Satellite Paradox: K+U=E

In summary, if the orbital radius of a satellite changes by a small amount, the resulting change in the total energy is -μ/2r.
  • #1
F.Turner
10
0
1.
{This paradox denotes the fact that a satellite in a near circular orbit suffers an increase in velocity when subject to a drag force.}

The Specific energy of the satellite is K + U = E where K =v^2/2 is specific kinetic energy. U= -u/r is specific energy and E is specific total energy. If satellite in circular orbit then u=(v^2)*r.

If orbital radius changes by a small amount dr, what is the resulting change dE of the total energy?

3. The Attempt at a Solution

What I figured to do is take derivative with respect to r but if I do that it will cancel out the distance completely when I plug in the u into the E equation. I I'm not sure if that's the approach I should take. Or should I take derivative with respect to time, because as time changes distance also changes. Not to sure...
 
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  • #2
F.Turner said:
...
What I figured to do is take derivative with respect to r but if I do that it will cancel out the distance completely when I plug in the u into the E equation...

Can you show what you mean by this?
 
  • #3
So if I take the derivative of the specific total energy E with respect to the distance value r something like this:

dE/dr = -[tex]\mu[/tex]/2r --> i plug in the value of [tex]\mu[/tex] = (v^2)*r

it turns into -(v^2)*r/2r the r will cancel in this case...(or maybe i should leave one alone and continue to eval the derivative...

The other case would be to take dE/dt to see the change in dr.
 
  • #4
This is paradoxically true -- assuming the drag force is small, of course.

What is the total energy for a circular orbit as a function of orbital radius?
 
  • #5
That is what I show up there its E=-[tex]\mu[/tex]/2r where [tex]\mu[/tex] = (v^2)r
 
  • #6
If -μ/r has dimensions of energy, then how can dE/dr = -μ/2r? The dimensions don't match.
 
  • #7
Ok then that would mean dE/dr wouldn't work so, it has to be dE/dt in order to get my answer; so in order to take the derivative with respect to time I would have to plug in v=d/t into the equation right...
 
  • #8
dE/dr is the correct approach.
 
  • #9
Okay thanks I got it
 

1) What is the Satellite Paradox?

The Satellite Paradox, also known as the Twin Paradox, is a thought experiment in special relativity that deals with the concept of time dilation. It involves two twins, one who stays on Earth and one who travels at high speeds in a spacecraft, and explores the idea that time can pass at different rates for each twin.

2) How does K+U=E relate to the Satellite Paradox?

K+U=E is an equation that represents the key components of the Satellite Paradox: Kinematics (K), Uniform motion (U), and Energy (E). These concepts are essential in understanding the time dilation effect in the thought experiment.

3) What is the significance of solving the Satellite Paradox?

Solving the Satellite Paradox can help us better understand the principles of special relativity and how time can be affected by motion. It also has practical applications in fields such as space travel and GPS technology, where precise time measurements are crucial.

4) How is the Satellite Paradox solved?

The Satellite Paradox is solved using the principles of special relativity, particularly the time dilation formula. By considering the different reference frames of the twins and the effects of their relative velocities, it can be shown that the twin who travels at high speeds will experience less time than the twin who remains on Earth.

5) Are there any real-life examples of the Satellite Paradox?

Yes, there are real-life examples of the Satellite Paradox. One of the most well-known examples is the use of atomic clocks on GPS satellites. Due to their high velocities in orbit, the clocks on the satellites run at a slightly slower rate than clocks on Earth, causing a time difference that must be accounted for in GPS calculations.

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