A train is traveling in the -x direction at 50 mph. A mosquito is traveling in the +x direction heading straight for the train at 1 mph. The mosquito strikes the train. During the collision the mosquito's velocity changes from + 1 mph to - 50 mph. At some point the velocity of the mosquito must be zero. Since the mosquito is in contact with the train, the velocity of the train must also be zero. What is wrong with this picture?

A 450 Ton train is NOT going to be slowed by a 1 gram mossie, so NO change in velocity at all.

This is interesting, this is clearly incorrect, but I don't know how to express it in words.

I think the answer is that the bug is not rigidly coupled to the train, so the bug may be moving 0, but the train will still be moving through it.

I don't know physics, but is this train in a vacuum? If not, isn't the air it's colliding with offering almost as much resistance as the mosquito and it's a lot bigger?
(Please don't negative me; as I said I don't know physics)

I think that the bug will feel differently as it splats against the train. It's about as coupled to the train as you can get! Velocity is a vector. The bug's velocity goes from +1 mph to -50 mph, it has to go through zero.

I'll give you a hint. It has to do with infinitesimals.

Let's assume the train is in a vacuum. It doesn't change the problem.

Also, what about mass? If I jump onto the ground, I'm shoving the planet backwards, right? How is the mosquito any different whether he splats, bounces off, or is coupled rigidly as a cyborg bug?

Sorry Ben, not true at all, there will be a change of velocity

:o
Bit harsh there!
True but so infinitesimally small as to be negligible..

You are missing the point. At some point the bug's velocity is zero. It is in contact with the train. Isn't the train's velocity also zero?

You are missing the point. At some point the bug's velocity is zero. It is in contact with the train. Isn't the train's velocity also zero?

As the bugs velocity is changing from +1 to -50 then it would indeed pass through zero, due to the change of direction.

Whereas the train would only decrease an INCREDIBLY SMALL amount and continue in the same direction. To all intense and purpose the velocity would still be +50 (or -50 in relation to the mossie)

However, at the small point of impact, the front of the train will absorb the momentum of the mosquito and be elastically compressed. The same goes for the front of the mosquito, which will absorb the momentum of the train, though it lacks the compressive strength of the metal or glass at the front of the train, and so it will be compressed far beyond the elastic limits of its exoskeleton. (Splat!) During the collision, the outermost molecules of the train will, indeed, be slowed. However, assuming that the mass of those molecules (and forces holding them in place - i.e. material stiffness) is on the same order as those in the mosquito's face, the outermost ones will be respectively accelerated to the average velocity of 24.5 mph. The rapid acceleration of the mosquito's face from -1 mph to 24.5 mph and the front of the train from 50 mph down to 24.5 mph (and eventually back up to 49.9999999999999999999999 mph for both) is facilitated by the spring-like Van der Waals forces among the molecules (and hydrogen bonds, etc. etc.). Hence, even on a molecular scale, the train and the mosquito aren't truly coupled.

So not only was the train, as macroscopically observed by us, basically unaffected by the collision, it's also unlikely that any part of the train ever slowed to 0 mph during the collision.

Woops, I didn't mean to start all that with "However". LOL! You can tell I deleted a sentence right before.

Maybe I don't understand the question, but I don't see any argument that because the bug's velocity is 0 the train's must also be zero at the same time. That would violate conservation of momentum and conservation of energy principles. From conservation of momentum - at the instant the bug is at 0 MPH you have:


m_{bug} \times v1_{bug} + m_{train} \times v1_{train} = m_{bug}\times 0 + m_{train} \times v2_{train}\\
v_2{train} = \frac {m_{bug} } {m_{train}} v1_{bug + v1_{train}


Hence at the point when the bug's velociity is zreo the train's veocity is decreased by only a very small amount:


\Delta v_{train} = \frac {m_{bug}} {m_{train}} v1_{bug}


The only way the train could be momentarily at 0 MPH is if the initial momentum of the bug equaled or exceeded the initial momentumn of the train, which is clearly not the case here.

Jcaron2 - your argument about the bug being compressed is off the mark. Even if both the tran and bug were made from incompressible materials (no splat, but a truly elastic collision) there is no argument that the train is ever slowed by more than an infintessimal amount.

I'm out of my element but am curious - does any of the bug's velocity go off to the sides if some of him goes off to the sides?

I'm out of my element but am curious - does any of the bug's velocity go off to the sides if some of him goes off to the sides?

Yup :)

Okey everyone. I'll give you once last hint. First, it's not a physics problem, it's more a calculus problem. This should give it away. It has to do with velocity being a change in distance with respect to time. Anyone got it now?

Ebaines, I think the point of this brainteaser was to introduce an apparent paradox. It appears macroscopically that the train and mosquito are instantly coupled (i.e. absolutely NO elasticity to the collision). The question then argues (wrongly in an attempt to trick the reader) that this would mean that upon their initial collision, the train and the mosquito are instantly traveling at the same velocity from that point forward. Yet somehow in the time after the initial collision the bug gets from 1 mph to -50 mph, which means it MUST pass the 0 mph mark, which means the train must also have done so. Of course this is ludicrous (hence the seeming paradox and the point of the riddle), as it violates the conservation of momentum, energy, and plain ol' common sense.

The resolution is simply that there's no such thing as a truly inelastic collision. If there was, the two objects would experience infinite acceleration, and in that case the change in velocity from 1 to -50 would take place in zero time so the train STILL wouldn't ever be slowed down more than an infinitesimal amount.

All in all, I think this was pretty weak as brainteasers go (though I'm sure the point is to make introductory physics students think about these very concepts - so to that end it works). So weak, in fact, that I think you never even saw it as a paradox because it's so obviously NOT one!

Actually I think the paradox here (if there is one at all) is that under classic physics the change in velocty of the bug and the train is instantaneous. Which means they both undergo infinite acceleration. And since F=ma, that means they each experiences an infinite force. So... does that mean that more energy is generated in this collision than in a 20 Megaton atomic bomb explosion? Hmmm...

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.

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. .

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.