The bee reverses direction (through zero obviously), the train does not. The momentum of the system remains constant throughout.
You lost me there, sorry.
All I’m saying is that if the bee reverses direction, then the bit of the train coupled to the bit of the bee reversing direction should be doing much the same thing.
If it doesn’t, it must be because the circumstances of coupling are sufficiently elastic and capacious that the bee bits have room - and time - to slow down and reverse while the train bits continue advancing without relent.
A point has no dimension.
That point at the end of my previous sentence? It’s not a point. It’s a dot, a fat point. You can’t draw a mathematical line, or a mathematical point, or a mathematical plane. A mathematical point is smaller than an electron, a mathematical line is narrower than the electron’s shadow 
(if an electron had a shadow, that is), a mathematical plane is thinner than anything we can build physically.
That point in time is a mathematical point. No dimension. It lasts exactly zero time.
Technically, her stinger. 
I don’t see why this must be true. Think about it carefully. You’ll see that a free-rolling train is accelerated backwards by the collision, but it keeps moving forward, thus not reversing direction.
To expand on my previous answer, it’s incorrect to say the rubber sheet will be traveling backwards at 83 mph. It will indeed travel backwards and reverse direction eventually but its initial speed will depend on the ratio of the masses involved.
I think it has to, assuming that the two masses stay in contact the whole time. Otherwise, the ball would be instantaneously decelerating to whatever the initial speed of travel the sheet goes. I suppose that things can happen “instantaneously” if the total time lapse is less than the Planck time, but otherwise I believe that instant things are frowned upon.
It’s not instantaneous because there is flexing but that does not support your conclusion that a microscopic piece of the train must stop. The acceleration of the microscopic piece of the train will cross zero, it’s velocity will not. Can anyone show otherwise?
If I’m wrong about this let the powers that be change my title from “Member” to “Another casualty in the war against ignorance”.
assuming a rigid train and bee (i.e. no splat factor), then why do we assume the bee must be at 0mph at some point. In this hypothetical situation, does velocity have to be continuous or can it go from +3 to -80 mph instantaeously.
It will accelerate to a velocity of -83mph from a velocity of +80mph. It seems rather difficult to achieve that without a velocity of 0 in there somewhere.
Any way you cut it, Pedro, if the bee passes through zero and is in contact with a part of the train while doing so, then that part of the train must also pass through zero. Otherwise you are talking about two things being in contact while not moving at the same speed, which (a) is clearly an oxymoron and (b) means that at the moment the bee is passing through zero it has nothing to accelerate it, since it is not in contact with anything, which means that it wouldn’t be passing through zero, which it is.
You thik that’s weird, try imagining a record spinning. The outer edge is moving faster than the the edge of the spindle hole.
CRAZY!
Following that logic, then the entire train must pass through zero. The bee is in contact with the “a” part of the train, which is in contact with the “b” (no puns, 'k?) part of the train, which is in contact with the “c” part of the train . . . Every part of the train is in contact with some part that’s in contact with some other part that eventually is in contact with the bee. Doesn’t seem quite reasonable, though, that the entire train, even momentarily, has zero speed.
Personally, I think this whole thing can be “solved” by looking at the atomic level. Pick a particular atom in the bee. When that atom does the pass through zero thing, it’s “in contact” with the train, but what does “in contact” mean at the atomic level? It means that there is a force from some of the train’s atoms that are acting on our bee atom. There is no physical connection, only a connection of force(s). So at that moment when the bee atom goes through zero (i.e. is stationary), the train’s atoms are still moving forward, and the distance between the train’s atoms and the bee atom gets smaller. But since that distance is just empty space, there’s no paradox, or even anything hard to grasp. And a moment after that, the bee atom (due to the forces acting on it) speeds up to 80 and is now moving happily along with the train atoms.
You know what flex means, right?
Following your logic the whole of a trampoline including the legs descends several feet every time the trampolinist hits the mat.
Simply, the part of the train that is in actual contact with the bee has to be going the same speed as the bee or they are not in contact. The other parts of the train are connected elastically (not very elastically, but elastically) to the part in contact with the bee.
Plus I never agreed to no puns. I plan on using them anytime it seems appropriate to do so.
As to the latter part of your post, I agree that if we are talking about a very hard small object striking a solid big object, the only elasticity might be at the level of inter-atomic forces. But with a large enough bee, and a soft enough train front, the train front itself would have to start to flex. There has to be (bee) flex involved: the question of whether that flex could merely be (bee) at the level of inter atomic forces is just one of degree.
I am shamed and unworthy of the light you all shine on me with your powerful, god-like intellects.
IANA physicist, but I’ve always wondered something similar about the bat contacting the baseball. Watching it in slow-mo, the bat “gives” at the moment of impact. But it seems like the ball has to stop moving toward the plate before it can move toward the centerfield bleachers.
It’s also interesting that the harder a pitcher throws, the farther the ball goes when hit, even though it would seem to take more energy to stop it and redirect it.
I guess the ball “stores” energy from the pitcher. The batter “adds” energy to the ball with the swing. Some of the energy is lost to friction when redirecting, but it’s mostly still in the ball. In tennis, they call it “pace”—under the right circumstances the ball goes faster and faster with each volley.
A bee isn’t very solid, doesn’t have much mass, but parts of it probably bounce off the train at a high rate of speed.
So, the conundrum remains unaswered. The whole flex issue, fascinating though it is does not give the answer in itself, but merely descibes another phenomenon that happens at the same time. I really loved the fact that at some point the bee is actually travelling faster than the train because of the elasticity/flex issue. However I remain unenlightened on the base question of two objects firmly connected so that they become one object with both having a different velocity at a single point in time. It’s like claiming that the back of the train is going faster/slower than the front. Perhaps the dimensionless point in time lies at the heart of this. Since that point has no dimension and therefore doesn’t really exist it could be that anything happening in that black hole doesn’t have to make sense !!
Well, the bee squashes up a lot, so I’m not convinced that it ever travels faster than the train. But if you had a steel ball bearing about the same size (volume or mass, it doesn’t really matter much) hitting the steel front of the train, it would bounce off the front, travelling at roughly twice the speed of the train. (The energy for this coming from the train slowing just just a tiny amount).
If there were no elasticity in the coupling, this would not be possible - that was my point.