Determining the velocity for a radial impact on an orbiting satellite

Question

Say a planet P and a satellite S (the size of a moon) system exists and the orbit of S around P is circular.

To make the satellite S crash into P, I can either slam an S sized comet C opposite to its tangential velocity and kill the momentum or slam the comet C parallel to the radial velocity at enough velocity that the orbit becomes too elliptic and S crashes into C after a time T.

I am having trouble understanding the physics of the radial impact though. I know from intuition and some orbital mechanics that the orbit of S would go elliptical and change direction over T. But I couldn't get these things straight.

How should I determine the required impact velocity to make the S crash into P in the case of radial impact? (For tangential impact, with the orbital velocity, I can determine the Kinetic Energy I should impact with. But what about radial impact?)

Answer 1

First, we can calculate the speed required for a satellite S to orbit the planet P $$F(r)=\frac {GMm}{r^2}=\frac {mv^2}{r}\implies v=\sqrt {\frac {GM}{r}}$$ Thus if speed will be smaller than the square root of the mass of the planet times gravity constant divided by distance from planet to satellite, the orbiting object will fall down.

If the radial impact happens, we can assume (though we are stretching are assumptions here), that the speed of orbiting object stays the same. Note however, that the r term get's smaller, thus in order for an object to keep on it's orbit, it has to have bigger speed.

The conclusion is obvious.