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?)