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Apparently Boscovich's argument shows that force must act at a distance:

But how did the velocity of the faster body come to be reduced from 12 to 9, and that of the slower body increased from 6 to 9? Clearly, the time interval for the change in velocities cannot be zero, for then, argued Boscovich, the instantaneous change in speed would violate the law of continuity. Furthermore, we would have to say that at the moment of impact, the speed of one body is simultaneously 12 and 9, which is patently absurd.

It is therefore necessary for the change in speed to take place in a small, yet finite, amount of time. But with this assumption, we arrive at yet another contradiction. Suppose, for example, that after a small interval of time, the speed of the faster body is 11, and that of the slower body is 7. But this would mean that they are not moving at the same velocity, and the front surface of the faster body would advance through the rear surface of the slower body, which is impossible because we have assumed that the bodies are impenetrable

Wouldn't be possible to explain this situation by discussing how the body is compressible instead?

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the problem with those arguments is that the description of collisions based solely on Newtonian mechanics is a good approximation that gives good results in terms of speeds before of after the collision, but if you try to force them beyond the intended use, they could lead to ridiculous (or contradictory) conclusions.
What actually happen in a real collision, is that the forces between the two bodies are electrical, they are the result of repulsion between the electrons of the atoms at the edge of both bodies. The electric force is a long range distance, so the interaction between the two colliding bodies starts quite before they enter in contact (actually they never enter into physical contact, unless there is a chemical reaction, because the repulsive forces will become strong enough to stop the bodies from actually "touching", I mean, if you had a very powerful zoom, you'll see that they never physically touch. this of course happen in a very small space and time scale, and macroscopically it is ok to idealize and assume that they actually touch each other to describe the results of an experiment. But to use that simplification to make arguments about what actually happens at the microscale, you will end up with those contradictions

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Indeed that is what happens, but how does Borscovish's argument prove this is the case, rather than just compressibility –  Casebash Nov 29 '12 at 23:13
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you are right, a small deformation of the objects as they enter into contact is another plausible explanation to the apparent paradox (which arises from assuming totally rigid objects). Actually, the cloud of electrons deforms as they approach each other, so in this sense the compressibility explanation is not only another alternative, but one of the aspects of what actually happen –  julian fernandez Nov 29 '12 at 23:22
    
@Casebash - It is possible to imagine two perfectly incompressible (and non-deformable) bodies colliding against each other. I suspect it was such a perfect scenario that led Borscovish to his paradox. –  Kitchi Dec 9 '12 at 15:43
    
I am not that sure you can imagine that (I mean, you can assume it, but imagine it it is a bit different!) just remember what Kant thought about euclidean geometry –  julian fernandez Jan 27 '13 at 4:45
    
@julianfernandez: The forces are not primarily electrical, but due to the increase in energy when the electron wavefunction clouds begin to overlap, due to the Pauli exclusion. This is obvious if you consider any sort of simple model of the situation, or even a complicated model, yet still everyone gets it wrong. –  Ron Maimon Aug 23 '13 at 1:01

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