If we take a satellite rotating around the earth, then the earth applies centripetal force to the satellite, but what if the satellite suddenly looses kinetic energy, maybe because of collision, the point is that now it goes slower. The gravitational force is the same, but the velocity is less, so does that mean that the satellite aside from rotating, will gain velocity towards earth until the gravitational force is just enough not to accelerate it towards the planet but keep it in uniform circular motion? If so, then the satellite, even when the gravitational force will be just enough, should keep it's velocity, and keep going towards the earth, and the distance should be smaller, and so the gravitational force should increase and the satellite will keep accelerating towards the earth and eventually will crash into the earth (unless it will burn on the way). By this logic the earth should accelerate objects towards itself with acceleration greater than 9.81 because we have to take into account that we all rotate around earth. how can we calculate that acceleration? And shouldn't then even minimal imperfections in the speed of satellites cause them to crash down to earth?
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2 Answers
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The satellite will indeed accelerate, but instead of falling to earth, it will form a thinner and longer orbit. And concerning the acceleration on earth, it is indeed changing due to centripetal acceleration, and the change is the smallest on the poles.
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A satellite colliding with a stationary object (in the non-rotating frame of the Earth) would be equivalent to a "retrograde burn".
For a satellite originally moving in a circular orbit, the resulting orbit would be elliptical (thinner), with a smaller semimajor axis. The collision point becomes the apoapsis (highest point) of the new orbit. Even though the velocity of the satellite is reduced by the collision, the time taken to complete an orbit would be reduced.
