Collision paradox Suppose an insect travelling south at some velocity, collides with a train travelling north at some other (opposite) velocity.
The insect hits the front of the train, where it splats to an unfortunate demise. However, at some point during the impact, surely the insect - upon becoming affixed to the front of the train - has to reverse its direction in order to continue in the same direction and speed as the train (which never stops or slows, and is unaffected by the event).
If so, doesn't this also mean that at some instant(s) in time the insect is travelling at 0mph (as it transitions from travelling south to north), while it is continually 'connected' to the train, which never stops?
How is this possible, or where am I going wrong with this logic?
edit:
I would like to revise the question to ask specifically about each atom/particle(/whatever) as it hits the train; what happens at this scale? Do we get squishy quarks? :)
 A: Yes, the insect does actually travel at 0mph when it reverses direction. The critical aspect that you are missing is that the time that it is not moving is infinitely short. Actually, I would think that the head stops, then the thorax and finally the tail as it is squished up against the train.
A: Both the insect and the train window are deformable.  Microscopically so, but deformable nonetheless.  Because of that fact, the insect slows continuously to zero, reverses direction, and then speeds up in the direction the train is going.  To our human perception this happens imperceptibly fast.
A: This is an interesting paradox. But it must have a solution, so let's try and find something that is adequate to our everyday experience: the train must not, obviously, stop.
I think the reason why this problem is startling at first is that it refers to an insect. When we think of an insect, we think of something with almost only two dimensions. But an insect, even a small one, has volume. 
So, if, instead of an insect, we imagine a cube, it is easier to imagine a scenario where the front of the cube (the one that faces the train) first approaches the train and then gets squeezed and deformed by the front of the train, while the other side of the cube is, for a very short period of time, at rest.
This is analogous to what happens with the insect.
A: I propose my solution to this paradox. Well let us assume the insect is travelling at velocity $v$ and train travelling towards the insect at speed $v_1$ (not in insects frame of reference, but trains frame of reference). 
That said, now the insect hits the train and as a result will decelerate at an finite amount of time $t$ as it is impossible to decelerate at $0$ seconds (as soon as insect is in contact with train) in ANY frame of reference because it would imply $\infty$ energy because of the equation of acceleration\deceleration $\frac{(0 - v)}{0}$ for example let us take $v$ to be $10$ m/s therefore it gives us an answer of $\infty$ as $\frac{-10}{0} = \infty$ and any physicist recognizes if an $\infty$ arises in any arbitrary system, there is an flaw in design of system and I used this and deduced if such a case exists conservation laws (energy conservation in specific) are broken with our pre-existing knowledge of basic physics we can deduce that the insect must have decelerated at some finite amount of time that is $t\ne 0$ that said we can safely say such an environment cannot exist. This suggests either the train has deformed very slightly or the insect has or a bit of both in order to obey the elegant conservation laws of the universe. 
That mathematical proof, suggests this thought experiment is not valid in real-world.
Now, keep that in mind such a design of a train may be impossible as Theory of Relativity tells us that the observer if changes his velocity in respect to the train the train may look as if it is slowing-down or speeding-up  depending on the observers change in velocity in respect to the frame of reference, so we can say such a object cannot exist as you have clearly stated the train may not be changed in velocity yet in observers frame of reference it has which shows an problem with the "ideal" conditions set in your thought experiment. 
