Take a hypothetical object with an infinite density (purely so that a particle can get incredibly close to it). Furthermore imagine that this object has no close range repulsive force, or in other words, particles can pass right through it. The only thing is does is exert a standard gravitational force. Now, if we drop a particle in the vicinity of this mass, it will accelerate towards it as always. However, due to the infinite density (r = 0) the force will eventually become infinite (F = GMm/r^2 and r=0) right? Now, I am fully aware of the implications of special relativity here but I would like to discuss this from a purely classical and non-relativistic perspective, it's a thought experiment, nothing more. The particle would feel an infinite acceleration and by taking the derivative of the acceleration function it is clear that the velocity is also inversely proportional to the distance, r. So it seems like the particle would reach and infinite velocity, BUT surely it would take an infinite amount of time for anything to reach and infinite speed...? So what really happens? In a purely classical, non-relativistic setting. Furthermore, what happens to the particle after it passes through the object? Does it slow down again and achieve a harmonic motion? Or does it fly through the object so fast that it just keeps going? Normally we don't need to worry about these things because celestial bodies like the Earth have a surface, and sooner or later any particle falling towards it would hit the surface. What happens when there is no surface and all the mass is concentrated at a point?