Reply to Comment on 'Forbidden collisions'

Rizcallah's Comment [1] refers to a recent paper [2] pointing out that solutions to one-dimensional collision problems can be found in which one body overtakes another, which physically does not make sense. The Comment offers several interesting points:

• 1.  
  The issue raised in [2] (how objects can overtake other objects during one-dimensional collisions) has very little to do with the penetrability of matter and much more with the incompleteness of interaction, in the sense that the given velocities are not final and the collision is ongoing.
• 2.  
  Using a centre-of-mass (CM) reference frame simplifies the criteria for forbidden collisions: only those in which both objects approach the CM before collision, but move away from it afterwards, are acceptable, legitimate collisions.
• 3.  
  There are simple ways of modelling collisions in which one object overtakes another, without actually going through it. A block sliding past a platform, but with friction between them, is given as an example.

To support point 1, the Comment reinterprets a problem given in [2], in which a metal object travelling to the right catches up with a slower, heavier stone also travelling to the right. After collision the final velocity of the stone (to the right) is given. The problem asks for the final velocity of the metal object, which turns out to be also to the right, but faster. Thus, the metal object appears to have gone through the stone. The Comment reframes the condition called 'after collision' and 'final velocity' in the problem as if it were an intermediate step in the process of an elastic collision in which the metal and stone do not exchange positions. While this is an interesting observation that may put some minds at ease, it completely misses the point of the original paper, which is precisely that if the condition called 'final' is final, there is still something very wrong going on even though the collision obeys momentum conservation. In the example above, the metal object will keep moving to the right faster than the stone after this condition has been reached, and their interaction will be over. Creating a fictional context does not fix the problem brought up in [2].

Regarding point 2, the condition stated in the Comment is equivalent to impenetrability, but it is formulated within a different reference frame, and certainly more elegant. It works for the catch-up-and-overtake scenario, but it does not appear useful for head-on-and-swap collisions.

Point 3 is illustrated by a clever Deus ex machina which tries to work around an overtaking collision without the need to penetrate matter. While the author deserves credit, he has ignored the two caveats in the original paper. The first is that this is no longer a true one-dimensional (single-axis) collision. The collisional force (friction) and the lines of motion of the block and platform are in the same direction but on different axes. The friction forces impart angular momentum with respect to the CMs of each object; this makes the motion two-dimensional. The second warning is that precisely in order to preserve the apparent one-dimensionality of the motion, physical constraints will be needed to eliminate the additional, unwelcome angular motion and hence will violate the most essential condition for conservation of momentum: no external forces.

In conclusion, the points brought up in the Comment are interesting and creative, and force a deeper look into the deceivingly simple phenomenon of one-dimensional mechanical collisions. Nevertheless, the arguments in this Reply support the view that forbidden collisions are still a problem that cannot easily be dismissed.