# The puzzle of the bee and the train

There is an issue I have been puzzling over for some time now, without success. True it’s not going to change the world but then, what does? That’s not the question by the way! The problem is, a bee flying West collides with a train heading East. The bee is travelling at 3 mph and hitting the train will decrease that very rapidly. It then moves to travelling at 80 mph in the opposite direction (the speed of the train) - since it is now glued firmly to the front of the train. Now in moving from 3 mph West to 80 mph East at some point (when it hits the train) it’s speed has to hit 0 mph i.e. for the briefest period of time it is stationary. So here’s the problem – you have two bodies, connected to each other, one of which is stationary and the other is moving at 80 mph? This strikes me as somewhat illogical.

By the way, no bees were hurt in the making of this question.

The bee is not totally rigid. At the moment when the average speed of what remains of the bee is zero, some of the bee is still travelling forward, while the part of the bee that has come into contact with the train is moving backward at roughly the speed of the train. In other words, the bee is being rapidly squashed.

(I’m ignoring the non-rigidity of the train: the impact of the bee slightly deforms that part of the train in contact with the bee, but that deformation is probably not measurable.)

No need to worry about that. As the train and bee approach each other, they will first be one meter apart, then half a meter, then a quarter and so on. It will always take some finite time to close the half the ramaining the distance, and then some more time to close half of what is left after that … ad infinitum. Infinite times finite is still infinite, so it’s impossible for the bee and the train to collide in the first place.

Now, play nice with the guest.

What is the last thing to go through the bee’s mind when he hits the train?

His stinger.

I thought this was going to be the one about the bee flying back and forth between two trains.

**Giles **gives us the real-world answer. If you, however, assume an infinitely rigid bee and train, the bee is at 0 MPH only for a point in time. A point in time is a little like a point in space; it has no duration. However, the other problem you have in this idealized situation is that the bee manages to change its velocity vector by 180[sup]o[/sup] and 83 MPH at a point in time, that is, without slowing down first or speeding up afterwards. Which, when you do the math, results in infinite acceleration, so clearly the idealized situation can’t happen in this universe.

This is exactly why I had to stop working on my Bee Cannon.

Me too.

Not that this question hasn’t already been answered, but the reason it’s confusing is because it’s so precisely phrased and uses somewhat odd objects.

A substantially similar question would be: “You have a soccer ball. You drop it toward the floor, but at the last minute give it a swift kick straight upward. Now in changing from a downward motion to an upward motion, at some point the ball’s speed has to hit 0 mph i.e. for the briefest period of time it is stationary. So here’s the problem – you have two bodies (a foot and a ball), connected to each other, one of which is stationary and the other is moving upward. How can this be?”

The answer to this question should be more intuitively obvious than the bee question, I think.

At the moment of impact the bee is going in one direction while the train is going in the opposite. So you would have two bodies joined and traveling in opposite directions.

In reality this is not true. You see, parts of the bee have already reversed direction while parts of the bee are still moving in the original direction. A phenomenon known in physics as a “splat”.

I think some of the posters have come close to answering this but to rephrase.
Idealised objects are infinitely rigid, bounce without deformation instantly and hence do through infinite accelleration doing so. However no real object can do this without breaking. Real objects deform, even if ever so slightly, so that infinite accelleration is avoided. So your bee deforms as it hits and “gradually” changes direction over a millisecond or two.

Why worry? At any instant the bee is not moving. Consequently, it cannot ever move. Ditto the train.

Incidentally, the “acceleration wave” passes through the bee at exactly the speed of sound in apê, UIVMM.

The front of the train deforms as the bee hits it, such that the point of initial contact decreases in velocity from 80 mph to 0 mph before bouncing back. The total time lapse is so small that the amount of deformation is minuscule and probably entirely unnoticeable.

Can’t you just use the impulse-momentum theorum to figure out the time? Give the train and bee a mass and figure out how long the train’s force needs to be applied before stopping the bee.

Snort!

Not sure I understand the concept of a point in time having no duration. Intuitively I feel that any point shoud have some dimension. Whether it does or not is key to the overall question because it is at that single point in time that the paradox of something stationary attached to something moving exists.

No, the key to it is that nothing involved is absolutely rigid.

At the time the stopped bee is in contact with an area at the front of the train, that area is also stopped because it is flexing microscopically backwards compared to the rest of the train.

Contemplate **zut’s **example. Or, if it helps, imagine something even more more extreme than a soccer ball and a foot.

Imagine instead of a bee that a baseball is travelling east, and hits a train travelling west with a really soft trampoline on the front. The baseball hits the trampoline, and stretches the trampoline mat east, against the direction of the train. At one particular instant, the baseball is stationary as it transitions from travelling east to travelling west. At that moment, that part of the trampoline mat which is in contact with the baseball is also stationary, but it is joined to the rest of the mat and thence to the rest of the train by an elastic membrane. There is consequently no paradox in the momentarily stationary baseball being attached to the moving train.

The bee/train example scenario is precisely the same, it’s just that the flex in the front of the train is many, many orders of magnitude smaller, to an extent that makes it hard to imagine.

This one took me a while to picture and grasp the concept of non-rigid objects.
Even if the front of the train has a cast iron plate mounted to the front of it and the object is not a bee but a stainless steel ball bearing there is still deformation in both the ball bearing and the iron plate.
An extreme close up view will show individual bits of the bearing and the surface of the plate accelerating, decelerating, and even stopping at various points in time.
Because the small deformation in the iron plate is whats changing velocity (not the entire train) the rest of the train moves along at an unchanged constant velocity.

Your “intuition” and your “feeling” are irrelevant. A point, by definition, has no dimension. Doesn’t matter whether it’s time or space.

And a point (in time or in space) is a mathematical abstraction, which cannot be directly observed. Anything which can be observed has some duration in time, or extension in space. Points are useful objects to construct physical theories around, but no more, even if in our theories we treat them as really observable things.