[Dr.Jim]

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?

[Curlyben]

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

[Capuchin]

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

[Capuchin]

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.

[joypulv]

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)

[Dr.Jim]

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.

[Dr.Jim]

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

[Dr.Jim]

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

[joypulv]

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?

[Curlyben]

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..

[Dr.Jim]

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?

[Curlyben]

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)

[jcaron2]

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.

[jcaron2]

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

[ebaines]

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:



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



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.

[joypulv]

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?

[Capuchin]

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 :)

[Dr.Jim]

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?

[jcaron2]

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!

[ebaines]

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...