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...