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To be more precise consider the following situation: Suppose a satellite is orbiting at a distance of $r_1$ from the centre of the earth. After some time, rockets are fired such that the new distance $r_2$ is less than $r_1$. I computed the change in the total energy and it came out to be negative. The question is to state and explain whether the rocket on the satellite is fired so as to produce a force in the direction of, or in the opposite direction, to the direction of motion (of the satellite).

My reasoning was as follows: To move to a lower orbit would mean that the PE must decrease. This would imply that the KE should have increased. The only way we can increase the KE is by having a force which is in the direction of the motion of the satellite. However, this answer is wrong. Please highlight the flaw in my reasoning and also present an argument that uses the fact the change in total energy of a satellite is negative...

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  • $\begingroup$ "The only way we can increase the KE is by having a force which is in the direction of the motion of the satellite." $\endgroup$
    – Declan
    Apr 13, 2016 at 4:22
  • $\begingroup$ Sorry, I pressed enter too quickly. "Please highlight the flaw in my reasoning" $\endgroup$
    – Declan
    Apr 13, 2016 at 4:23
  • $\begingroup$ For an external force in the direction of motion (engine is external), the circular orbit will becoming elliptical, or hyperbolic. $\endgroup$
    – Declan
    Apr 13, 2016 at 4:25

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The final orbit has a smaller radius so we know the final PE must be less (more negative) and the final KE must be greater (moving faster in a smaller orbit) but the final total energy must be less (more negative), since $E_{total}=-\frac{1}{2}G\frac{Mm_{e}}{r}$. This means the satellite, or more correctly, the Earth-satellite system, has to lose energy. We need to apply our force in the opposite direction to displacement to do negative work.

So, how does the satellite end up moving faster in the new orbit? As it moves down towards the Earth the force of gravity (or at least a component of it) is in the direction of its displacement speeding it up. You can think of the satellite as falling into its new orbit.

It's all nicely explained by Rhett Allain in this Wired article.

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