Four Snails Traveling on a Plane -- Do they cross paths?

[kuruman]

Why to consider same speed of all snails?

It does not have to be the same. See post #29.

 

[mital]

The occurrence of sixth collision will depend on the velocities of the snails since it may happen that although their paths intersect, they do not meet as they can be at different positions at that time. Also, another case may arise when t=0 is chosen such that the two of the paths have crossed at t<0. Then only 5 collisions will occur(t>0).

 

[haruspex]

The occurrence of sixth collision will depend on the velocities of the snails since it may happen that although their paths intersect, they do not meet as they can be at different positions at that time. Also, another case may arise when t=0 is chosen such that the two of the paths have crossed at t<0. Then only 5 collisions will occur(t>0).

It depends on each snail having a constant velocity. As proved in post #29, it does not depend on all having the same speed.

 

[TSny]

Unless I'm overlooking something, I think @Steve4Physics has a nice approach in post #21.

Given that the 5 encounters AB, AC, AD, BC, and BD occur, we want to prove that the sixth encounter CD must occur. Steve4Physics has essentially shown that in the frame of reference of snail A, snails B, C, and D must move along the same straight line. To recap the reasoning, suppose that in the A-frame, B and C do not move along the same straight line:

The black dot is snail A at rest in this frame, the brown circle is B, and the blue circle is C. B encounters A at the instant shown. C cannot also encounter A at this instant since that would imply that A, B, and C occupy the same spatial point in the original reference frame at this instant. But we are given that no more than two snail paths can cross at any one point of the original frame. It should then be clear that for any speeds of B and C in the picture above, B and C can never encounter each other if they are moving along different lines in the A-frame. We conclude that the only way that B and C can encounter each other is for B and C to move along the same straight line in the A-frame. The same argument holds for B and D. Thus, all three snails B, C, and D move along the same line in the A-frame. (They don’t necessarily move in the same direction along the line.)

@jbriggs444 already pointed out in post #26 that nonparallel paths in the original frame can become parallel in the A-frame. We see that the nonparallel paths of B, C, and D in the original frame become the same line in the A-frame.

We can also conclude that in the A-frame the velocity vectors of B, C, and D must all be different. If two of the snails had the same velocity in the A-frame, then the two snails would have the same velocity in the original frame. This would imply parallel paths for these two snails in the original frame. However, the problem statement assumes that none of the paths are parallel in the original frame.

So, in the A-frame, snails B, C, and D move along the same straight line with different velocities. In particular, C and D move along the same line with different velocities. Hence C and D must have an encounter, which is what we wanted to show. We conclude that 5 encounters imply 6 encounters.