# Classical mechanics and the speed of a train-mosquito collision, when perfectly rigid bodies

This is all under the assumption that they are perfectly rigid bodies:

A train is moving at 300m/s. A mosquito is moving directly towards it, head-on, at 4m/s.

When the mosquito and the train collide, the mosquito is at 0m/s, and then changes direction, moving at 300m/s in the direction of the train (because it's stuck to the windscreen).

Surely this says that, in theory, because there is a period of time where the mosquito is at 0m/s, the train must also be at 0m/s. But the train surely doesn't stop moving, and a mosquito certainly couldn't stop a train, even for the smallest amount of time.

1. Is this really what theory predicts? If not, where am I going wrong?
2. Is the theory flawed with this almost paradoxical situation?
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What makes you say "the train must also be at 0m/s"? This simply does not follow. Think about the effect the train has on the bug. Think about how you would define velocity for a body as it is distorted by a sudden impact (meaning that not every part of it is moving at the same speed). Recall that even "contact" forces take place over some non-zero distance. –  dmckee Dec 11 '12 at 21:16
Comment to the question(v3): Now OP is considering the unrealistic/idealized situation of undeformed rigid bodies. In this idealized model, the velocity profiles of mosquito and train will be two discontinuous step functions, although the step in the train's velocity profile will be incredibly small. –  Qmechanic Dec 13 '12 at 6:34
What is puzzling you is quite close to the Motion Paradoxes of Zeno, rather than a real physics problem. The conclusion of the zero speed of the train is not correct, but anyway I think you should google for the greek philosopher Zeno. You will find his paradoxes interesting. –  Eduardo Guerras Valera Dec 16 '12 at 2:49

Right before the collision, the mosquito is moving at 4 m/s, right after the collision it is moving at -300 m/s. In between, it will be quickly accelerated, so yes, it will eventually be not moving for an instant. But the train and the mosquito velocities are only equal after the collision, not while it is happening. So there is nothing in theory that predicts the train stopping at all.

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Mosquitos, windshields and trains are NOT perfectly rigid bodies.. In fact, the mosquito will suffer a completely inelastic collision with the windshield and will get squished when it hits. So the mosquito center of mass will gradually decrease from 4m/s to 0m/s and then to -300m/s and all of this will happen over the time frame of the squishing of the mosquito. During this same short time the windshield may change from 300m/s to 299.99999m/s (I made up this particular number, of course, I don't know how many 9s are needed) and back to 300m/s. This happens because even though the windshield seems like it is perfectly rigid, it really does have an elastic modulus so the windshield will very slightly deform as it applies the necessary force to decellerate and accelerate the squashed mosquito to the speed of the train. So this is what happens in reality and there is no problem - a finite but large force will act over a short distance (the squishing of the mosquito) to reverse the velocity of the mosquito. The train never decreases to 0 velocity.

What would happen if the mosquito, the train and the windshield really are perfectly rigid bodies? Well in that case, the large force that acted over a short time and distance would become an INFINITELY large force acting over an INFINITESIMALLY small time and distance that will give infinite acceleration to the mosquito which will cause a discontinuous change in the velocity of the center of mass of the mosquito so it will not spend anytime at all at zero velocity. And of course the train will not decrease to 0 velocity at all.

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To clarify, it sounds like you are suggesting that:
1) There is an instant where the mosquito's velocity is 0, and:
2) Since the mosquito is stuck to the train's windshield...
It follows that that the train must also have an instant where its velocity is 0.

Of course, however, this is not really what theory predicts...

The discrepancy comes from the fact that as the mosquito hits the train, it is being smashed as well, and thus the velocity is never zero at every single part of the bug simultaneously. The mosquito is distorted (squished, in fact) as the train exerts its force in the opposite direction, which will cause the rapid deceleration of the mosquito. As this happens, the train's massive inertia rapidly reverses the bug's direction, but the leading edge of the train experiences only the practically insignificant force of the bug hitting its windshield and thus does not slow a significant amount.

To clarify, think about a car hitting a wall. The car initially has some non-zero velocity (say, $50 m/s$) and as it smashes into the wall (compacting the car) it decelerates until it is eventually at $0m/s$. Yet the wall does not move with the car even though they are in contact for the entire deceleration.

Hope this helps!

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The magnitude of the force on the mosquito and train are equal. The force on the mosquito, however, results in the dire consequences from its acceleration (change in velocity over change in time) which is not shared by the train. The force magnitude on the train is an equal and opposite reaction to the force from the squished mosquito. The magnitude of the equal force on train and mosquito, multiplied by the change in time during the event, would have to equal the mass of the train times its change in velocity from 300 to 0 to 300, in order for the train to stop. $F = ma$, the $m$ doesn't stand for mosquito.

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## protected by Qmechanic♦Sep 25 '13 at 18:10

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