Yet, as Newton and Leibniz first showed, it can be quite useful to embrace the idea that such things as velocity and acceleration need not be zero at a point, even when time is.
Not to worry. Johann Pachelbel successfully developed a marvelous D Canon.
RR
A previous thread more or less on this topic (If you ignore the differences between a fly and a bee)
Yes, this is true. The closest that the bee can ever get is 0.9999999… of the initial distance, and this will never equal one.
Uh-oh . . . are we opening up that can of worms again? (or in this case, bees)
And it’s at this point that half a bee, philisophically, must ipso facto half not be.
This gets me thinking about an issue that I’ve pondered for some time: namely that there is, in terms of time, only future and past. The present is but a point, and has no dimension. It only represents the dividing line between past and future. Do any physics or cosmology concepts that deal with time require the existence of a present, aside from a general reference to “now” or “currently”?
Why do you assume reality should be logical?
Ok, no offense, but all of you who are talking about rigidity, etc., have totally missed the point, and you are making a grevious error in the process.
Even assuming deformation of the train, the bee, etc., every bit of the bee at some point switches direction. At that moment, the velocity of that piece of bee goes from a vector headed east to a vector headed west. That piece of bee is for some point in time going to have a velocity of 0, attached to a train (and other bee parts) that is moving.
The point is, there is nothing paradoxical about this fact at all. Just because the bee piece is next to the train doesn’t mean that the bee piece is part of the train. So the bee piece is simply undergoing an acceleration different from the acceleration of the train (and the other bee parts). The only reason that we can take the bee as being part of the “train” after the splat is finished is because after the acceleration of the bee, the bee’s pieces parts end up having the same velocity as the train. Suddenly stop the train with sufficient decceleration, and you might well see the bee reacquire an identity different than the bee-train system.
Bee-train? No thanks, I’ll take the A-train.
There is, of course, a frame in which the Train comes to a complete stop.
It’s the first time in my life I’ve ever loved physics talk and discussion; here on this board. You guys crack me up.
No, it’s you who is missing the point.
The bee and the train don’t need to be melded into one or glued together and the bee doesn’t have to be “part of the train” for the situation described by the OP to be slightly interesting. The point is, at the moment the bee or a piece of it went from having a vector east to having a vector west it passed through a moment when it was stationary, and at that moment it was, and indeed must have been, *in contact with *the moving train. If it wasn’t in contact with the train there would have been nothing to accelerate it from moving east to moving west. Yet, the train was moving at an apparent constant speed but the bee or bee part or whatever was deccelerating/accelerating rapidly and indeed passing through an instant in which it was stationary.
Nor is the apparent difficulty resolved by pondering that the bee disintegrates: exactly the same apparent paradox would arise if we were talking about a single molecule.
The answer must and does lie in flex.
I’m not convinced by the flex argument. That would appear to do no more than explain the actual process where two bodies meet. It doesn’t address the central issue of a stationary object and a moving object being ‘one’ I do however like the argument put by DSYoung that the objects remain discrete and the fact that they happen to be touching at a moment in time is merely coincidental.
My sympathies to Panache 45 who views a point in time as having no dimension. For him there is no present - and possibly no existence…which is a whole new hive of bees.
Flex avoids having infinite acceleration, a physical impossibility. But flexing does not mean a microscopic piece of the train is actually stopped (just slowed down).
If you assume perfectly rigid bodies the bee is indeed “stopped” (has zero velocity) in a single point in time, without stopping the train. This does not break any physical law.
I have hit upon the difficulty here and the fallacy exposed is not so different from Zeno’s paradox.
In effect finding the instantaneous speed involves taking a limit, which requires a time interval. It does not make sense to talk about instantaneous speed without a time interval (this is unintuitive).
People equate a point in time with looking at a still picture but the difference is subtle. When looking at a still picture of a moving object, the object is indeed stopped. When “looking” at a point in time in an object’s trajectory in space-time, the object is not stopped, time is.
An example will hopefully make this obvious. Consider two perfectly rigid, identical snooker balls. Ball_1 is stopped and Ball_2 has speed V. They collide. We know from the law of conservation of linear momentum that Ball_2 will stop and Ball_1 will have speed V. This is not controversial. But if you look at the point in time where they collide, the balls are both stopped, violating the conservation of linear momentum law.
It does mean that a microscopic piece of the train was stopped.
Imagine a sheet of rubber stretched over a frame. I throw a ball at the rubber at 60mph. When the ball touches the rubber, it pushes it back at 60mph and is decelerated to 0mph as it hits the furthest point back.
If I tighten the rubber and throw it again, the amount that the rubber stretches back will be less, but it will still, for absolute certain, be accelerated backwards at first at the same velocity as the ball was traveling.
No matter how tense you make the rubber, it will always do just that. It doesn’t matter how massive the ball is or how fast it is going, there won’t be a difference except for the size of radial influence and the distance that is traveled backwards. So if you have a bee sized ball that is traveling 83 mph, and the rubber sheet is tensed to be as inflexible as metal, then yes on a microscopic scale, a small segment of the metal sheet will momentarily be traveling backwards at 83 mph.
I think I may see an issue with the question that lends it some very minor confusion. There is no distinction made as to the location of the observer. Whether or not the bee flexes or the train flexes is moot, both do. Whether or not the bee or a portion of the train is instantaniously motionless is moot, they both are. The difficulty I draw from this thread is:
If the observer is standing on the train, on the little platform right up front waiting for the yellow log to blow, then from their POV the train is motionless and the bee flys into it at 83 mph. Much like standing next to the trampoline, on a microscopic level the train flexes back initially at 83 mph but rapidly decelerating to 0 mph before flexing forward again. In the case of the bee hitting the train this flex may only be a few atoms deep and may last a tiny portion of time but it still happens.
Now move to the relative position of an observer on the side of the track. At the precise instant that the train’s flexion stops, just before the atoms bounce back whatever distance they are going to based of the elasticity of the train, the point in space that the deceleration occupies appears to be traveling by at 80 mph. Therefore, depending on the relative position of the observer the vector change either takes place at 0 mph or at 80 mph. I’m pretty sure uncertainty says it does both at the same time.
Interestingly, at some point in time the bee is actually moving faster than the train. As the atomic matrix flexes outward at a speed of x then the bee will be traveling along a vector in the direction of the train at a speed of x + 80 mph. When the flex reaches it’s final resting point both componants of the equation are moving along at the same speed and all of the componants of the bee are splattered across the front of the train.
That’s correct as far as it goes, with the caveats I mentioned before attached to the word “momentarily”.
If the very tense rubber sheet is traveling faster than the bee sized ball in the opposite direction and is very much more massive than the bee sized ball it will absolutely not do that. It will just slow down very slightly.
Parts of the system will have to reverse direction momentarily, no matter what, because the parts of the bee can’t reverse velocity without going through zero.
I guess it may be that those parts of the system that do this are not physical particles though - it could be that there is ample elasticity in the modes of interaction/contact between the bee and the train that they can take up the slack, permitting the bee’s particles to change direction without any of the train’s particles being in intimate enough contact so as to have to match them.
The whole conundrum as commonly posed, does really assume absolutely rigid objects - which of course cannot exist. If an object was infinitely rigid, applied forces would propagate through it instantaneously - i.e. faster than the speed of light.