There is no paradox here, and no laws of physics are violated. It requires an understanding of the difference between a finite change and an infinitesimally small change. We normally think of velocity as a finite change in distance with respect to time, but in fact that is not correct. If you remember from calculus that v = dx/dt, it is clear that both the mosquito and the train are at zero velocity for a distance in the limit in which the time is zero. For all practical purposes, neither the mosquito nor the train are ever at zero velocity. Obviously, the mosquito does not stop the train.

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Originally Posted by Dr.Jim

If you remember from calculus that v = dx/dt, it is clear that both the mosquito and the train are at zero velocity for a distance in the limit in which the time period is zero. .

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DrJim -If I follow you correctly your "paradox" is that since v = dx/dt, if Delta x is zero then v is 0, right? If that's it, the explanation is of course trivial - that the definition of a derivative like dx/dt involves the limit of what happens as delta t goes to 0, but does not concern itself with what happens if delta t is zero itself. But then if this is what you're getting at, why involve the mosquito? You could simply have said something like this: A train moves at 50 MPH, but since distance = rate times time then over a period of 0 seconds the train moves 0 feet. If it moves zero feet it must be stationary. Therefore a train that moves at 50MPH is stationary. Is this the same thing you're trying to get at?

Sooooo confuuuused. The mosquito is clearly at 0 (mean) velocity at some point in time, infinitesimal or not. Whereas the train clearly is never traveling at 0.

It's a paradox, allright. As Zeno pointed out some centuries ago, the problem is not that the mosquito's velocity is zero at some point in time, it's that the mosquito NEVER EVEN HITS THE TRAIN!! (It doesn't matter if Achilles and the tortoise are traveling in the same direction or in opposite directions. The argument is the same.)

Uuuuuuhhhh... yeah.

Sorry Dr. Jim. I can assure you that I have a thorough comprehension of simple derivatives and the idea of infinitesimal changes versus finite changes. I can also assure you that ebaines, Capuchin, and Dr. Bob do as well. I think they will all agree with me that at NO point is the derivative of the train's position (a.k.a. it's velocity) EVER zero. If you consider the fact that the train's speed was reduced by a finite (albeit extremely small) amount, then there will be a discontinuity in the velocity (implying infinite acceleration as we talked about above), but it will NOT be zero. I think ebaines said it pretty well above; of course the train will move zero distance in zero time, but that doesn't mean it has zero velocity (you can't compute velocity over a zero-length time because division by zero is ambiguous). That's precisely WHY we have to use the limit theorem for finite differences (in other words, the derivative).

That is "what is wrong with the picture" that I asked in the original problem. The mosquito is traveling at 1 mph (+x direction) before it hits the train and at 50 mph (-x direction) after the collision. One would assume that in order to do this, at some point it's velocity would have to be zero. But, if one understands that velocity is (delta x/delta t) in the limit that delta t goes to zero, neither the mosquito's nor the trains velocity are actually zero for any finite amount of time. It's not a paradox. The simple answer to the problem would have been, "Your statement that at some point the velocity of the mosquito must be zero" is incorrect. QED

By the way, the idea that the mosquito never hits the train is interesting. I wonder how the mosquito feels about that? I suppose I could have said that the train hits the mosquito, but I don't think that the mosquito would understand the difference.

But your statement that at some point the velocity of the mosquito must be zero IS correct. Just because the mosquito is accelerating and, therefore, never spends any finite amount of time at any one velocity, that doesn't mean it was never at that velocity. It was just there for an infinitesimal amount of time - that's the VERY DEFINITION of a "point" (as in "at some point its velocity would have to be zero). This assumes, of course, that the mosquito's change in velocity from +1mph to -50mph took some finite amount of time in total (like the time for his face to crush in). If we treat this purely mathematically and pretend that the change in velocity was instantaneous, then the velocity is discontinuous and the acceleration to both the mosquito and train are infinite. In that case, I can agree with you - the mosquito was never at 0 mph at any point. However, talking about the derivative in such a situation is meaningless, as it's undefined.

You just answered your own question. I did not say that the change in velocity from +1 mph to -50 mph was instantaneous. I said that in the limit that t goes to zero, the change in their position is infinitesimally small. Moreover, the mosquito is accelerating, you correctly point out that in the limit that t goes to zero the change in velocity is infinitesimally small. So, I suppose you can argue that the mosquito does stop the train for an infinitesimally small period. But now we are getting into semantics. You and others claim this is a paradox. I would appreciate your pointing out what the paradox is.

DrJim - this is really dragging on way too long, but I think what I, jcaron2, DrBob1, and Capuchin don't get is that you keep saying that since the mosquito's velocity is 0 at some point in time, then the train is also zero at that point in time. Why do you keep saying that?

Exactly!

I don't HAVE a question. I understand this just fine. If we agree that the mosquito did actually accelerate then it WAS at 0 mph for some point in time.

That in no way means the train was ever at 0 mph, so you're right. There is no paradox here (and please note that I always said it was an apparent paradox - meaning that the original question was trying to trick the reader into thinking it's a paradox which has to be rectified - not that it actually IS one). It was a blatantly wrong (and seemingly paradoxical) statement in your original question: "Since the mosquito is in contact with the train, the velocity of the train must also be zero." I was giving you the benefit of the doubt that you were providing a riddle. Apparently you actually believe the train's velocity went to zero in the limit as t approaches zero (as evidenced by your statement "If you remember from calculus that v = dx/dt, it is clear that both the mosquito and the train are at zero velocity for a distance in the limit in which the time period is zero.". Whiloe that statement is true for the mosquito, I can assure you that the train was never at zero velocity, not even for an infinitesimal amount of time.