Can anyone explain this?

Question

One of my kids came home from school with this problem from her maths teacher, and I can't think of a way to explain it :

A train is going west to east at 100mph
A fly is flying east to west at 5mph
The fly hits the trains windscreen
The movement of the fly therefore changes from east-west to west-east (in effect it changes direction 180°)
So there must be a moment when the fly's velocity is zero
That moment comes when the fly hits the train
So if the fly's velocity is zero when it's touching the train, then so must be the train's velocity
So the fly has stopped the train

This reminds me of that one where if you shoot an arrow at a tortoise it will never hit it because by the time the arrow has got to where the tortoise was, the tortoise has moved on a bit. I can't explain that one either!

Please help me get back some Dad-cred by explaining what's going on!

Answer 1

Watch a slow-mo of a tennis racket hitting a ball. The tennis racket as a whole is not stopped by the ball but the strings in the middle of the ball are stopped briefly, they then rebound and fling the ball on its way.

In the case of the fly hitting the train, the amount of the train which needs to "stop" in order to absorb the momentum of the fly is down to a very, very, very small area of glass a few atoms thick. If you looked with a microscope, you might find a very very small dent where the fly hit the train - though the glass will probably, like the tennis racket strings, have enough elasticity so that no dent is left.

Answer 2

Yes, word paradoxes can be entertaining, can't they, but that is what this is; it does not treat the physical reality of the situation.

I think that the fallacy is in thinking in terms of velocity alone.

Even if there is a moment when you could consider the fly's velocity to be zero, it does not follow that the train's velocity must be zero. The impact of the fly on the train will retard the train slightly (VERY slightly), but consider the energy of the trains motion, which is huge, and the energy of the fly's motion, which is miniscule. (I can't tell you what these energies may be off the top of my head, but perhaps if you ask that as a question, you may find someone who knows !).
Then subtract the fly's energy from the energy of the train, and it will tell you how much it will slow down the train -- obviously it cannot match the energy of the train, so it cannot retard if by very much, even if the fly were travelling at a million miles per hour.

Then again, remember that the fly is not a point object -- it has a definite size. First its head hits the train and gets splattered, then a microsecond later the front of its body, then a bit more later a bit more of its body, and so on, and so on, depending on how finely you slice the moments of time. So there is no ONE moment when the fly hits the train, it crumples gradually over a finite period, just as a car hitting the train would. The energy of its motion is gradually dissipated, first in crushing the body (of the car or the fly) and maybe in putting a dent in the windscreen, but it is not instantaneous.

Your second problem is a restatement of the rather more usual story of Achilles and the tortoise. Achilles proposes a race with the tortoise, but he graciously gives the tortoise a head of half the length of the course. When the race starts, Achilles first has to cover the first half of the course before he can overtake the tortoise - but the tortoise will have moved on a certain distance by then, naturally ; so Achilles then has to cover THAT disance before he can overtake ; but lo and behold, the tortoise has moved on again - so again Achilles has that distance to cover before he can overtake - but the tortoise has moved again ------ etc, etc. Conclusion : Achilles can never catch the tortoise !

The fallacy here is that the wording of the story masks the fact that the distances are being chopped into ever-smaller bits, and the story implies that these continual additions never add up to the total distance between Achilles and the tortoise.

Mathematics can be used to show, however, that a never-ending series of smaller and smaller numbers can, indeed add up to a finite quantity - and a surprisingly small quantity at that. So for example the series

1 + 1/2 + 1/4 + 1/8 + 1/16 + .......... etc

always comes to less than 2, however many terms you add on, and this is analagous to Achilles' problem - of course he will catch the tortoise, and very quickly. What the wording of the problem makes you forget is that each stage of the catching-up process takes a shorter and shorter time ; and if the number-series above represents the times which Achilles needs to cover the remaining distance between himself and the tortoise, then it all adds up to less than 2, however many stages you choose to chop it into !

I suggest you do a search for Zeno's Paradoxes, which will surely give you many references to the above problem, and many more of the same type.

Give the little blighter those to chew over !

Answer 3

There is a point when the AVERAGE speed of the fly is zero.
There is also a point where the leading edge of the fly is moving east while the trailing edge is still moving west. Depending on how squishy the fly is that change might happen as fast as the speed of sound in the medium of fly (or Newton's ball) or the fly might get compressed.

Relativity has nothing to do with it, but the nature of molecules might. The windshield might begin repelling the fly an infinitesmal time before the fly is actually in the same place as the windshield - the fly can turn around without every "touching" the windshield. But the tennis racket example given doesn't even make that necessary.

If a rubber ball were bouncing against an "incompressible" stationary surface there would be a point where the leading edge is against the stationary surface, as close as it's going to be, but the trailing edge is still going, compressing the ball. Not sure if that does it -- I suppose at the point of maximum compression there is a flattened ball that is, for an instant, absolutely (at the macro level) still, as still as a compressed and latched spring.

UPDATE: Relativity doesn't come in, but the concept necessary for it, frames of reference, does. Folks in the train think they are not moving. They see a ball coming at them at 105mph suddenly slowing down to 0 mph and then accelerating to 5mph. (Infinitesmally more than 5 mph because while the ball is in contact with the train it is getting energy from the train - this is how the "slingshot" works for deep solar system probes.)

There is a point where the ball and the train are going at the same speed (what's that theorem, that if a continuous function is at A and at B then at some point it has a value for every value between A and B ?) but that is when the ball is going 0 mph RELATIVE TO THE TRAIN which means 100 mph groundspeed.

(I've changed the fly to a ball. Balls bounce, flies stick.)

Answer 4

Why do assume the train's velocity is zero?

You have two forces here: that of the fly and that of the train. The sum of the forces before and after the collision is equal.

Force = mass times acceleration. The train decelerates ever so slightly, and the fly has its velocity go from its v to minus the v of the train in a hurry.

About shooting the arrow, sure the distance between the arrow and the tortoise keeps decreasing by 1/2 and 1/2 etc. but so does the time interval keep decreasing by 1/2, 1/2 etc. So if you do the math, you end up dividing zero (interval) by zero (time) which is meaningless. That's why the arrow catches up to the tortoise. People only focus on the decreasing space interval. They never consider the corresponding time interval. In other words, if the distance decreases by d, then 1/2 d then 1/4 d etc, so does the time decrease by t, 1/2 t, 1/4 t etc.

Answer 5

The train doesn't have zero velocity like the fly, because the mass of the train is so much more than the mass of the fly. The larger mass of the train means the small mass doesn't affect it's velocity. Whereas the fly's velocity is affected by the larger train.

As for the tortoise, the arrow's fast speed would mean it would reach the tortoise before the tortoise had enough time to move. The arrow is faster, which is why it is effective!!

Answer 6

The tortoise problem is solved by looking at the relative speed of tortoise and arrow. Suppose the arrow travels ten times faster than the tortoise and the tortoise is 100 metres away. By the time the arrow has travelled the 100 metres to where the tortoise was, the tortoise has moved 10 metres further on. By the time the arrow has travelled the next 10 metres, the tortoise has moved another metre and so on. Because space is always divisible it appears that the arrow never catches up with the tortoise and the distance travelled by the arrow is 111.111111 recurring metres. However this overlooks the fact that it is a continuously decreasing distance that closes. If you translate the decimal into a fraction it becomes obvious that the arrow does catch up with the tortoise after 111 1/9 metres.

Answer 7

it's all relative. when we are in a train then we would have surely observed the other train moving whilist ours was just at rest. this happens for the passnegers too on the second train.

similarily, when the fly hits the windscreen of the train the velocity of the fly may seem 0 at a point but the train would be moving. similarily, the people on the train would assume themselves at rest and take the speed of the fly to be something else.

this is true in all motion related situations. and is known as special theory of relativity. but if both the train and the fly were at zero speeds then how would the direction of the fly change?(no energy transfer has taken place since both objects haven't moved)!

Answer 8

Try to picture a slugger with a baseball bat hitting a home rum. Although the ball suddenly reverses direction the bat follows a smooth follow-through (never stopping). The ball didn't stop instantly either because it deformed (out of round) upon absorbing energy from the bat. It is true that the bat 'slowed' slightly upon hitting the ball (because it lost energy). The fly will also deform (splat!) and slow down the train in proportion to the mass and velocity of the fly with respect to the mass and (opposite) velocity of the train. But it is no contest (unless the 'fly' is a Boeing 747?).

Answer 9

If the fly accelerates from 5mph east to west (that's -5mph west to east) to 100mph west to east *instantaneously* then it was only at zero mph for such as small instant of time as to be insignificant. During that time the train would still have been travelling at 100mph.

You need to draw a graph showing the velocity of the fly to get the idea.

Answer 10

Well, the change of velocity of the fly is almost instantaneous, But the trains velocity will not change because the momentum of the fly can never have the energy to change the velocity of the train.