# Swimming pool and meteorite problem

As you kow, there is only one way to do logic problems, and that is democratically.

Suppose there is an Olympic sized swimming pool situated directly on the earth’s equator, running east to west in its length. Also, there is a second swimming pool of identical size, shape and orientation, but located on the Tropic of Cancer. If the probability of falling space debris is exactly equal around the globe, which one of the two swimming pools would you expect is more likely to catch more meteorites over a few million years?

Mintaka

I went with equal but feel there may be a catch

Oooh! I do feel a bit out of my depth - I must be in the deep end. Here goes anyway.

1. Do pools catch meteorites? I don’t know. Maybe fill it with rwenzori’s patent meteorite-catching mieliepap&kevlar compound.

2. I’m not sure about your assumption about probability - it’s not quite enough, as one needs to know direction. If true, then I suppose pools are equal. But…

3. Most matter in the solar system is in the plane of the planets or so, roughly, is it not? So most meteorites are coming in that plane at earth, roughly? If so…

4. The Tropic of Cancer pool represents a greater proportion of the latitude and thus is likely to trap more. But…

5. Because the meteorites coming at the Northern pool are coming in at an angle, they must pass through more atmosphere and are more likely to burn up before reaching the pool, therefore less being trapped.

So I don’t know. I do have the funny feeling that by this logic there must be a shitload of meteorites at the Poles. ??? ???

5. Because the meteorites coming at the Northern pool are coming in at an angle, they must [b]pass through more atmosphere [/b] and are more likely to burn up before reaching the pool, therefore less being trapped. (My bold)

But does it? Isn’t the atmosphere, in turn, thicker at the equator?

I was thinking that, with each rotation of the globe, the equatorial pool must sweep through a larger area compared to the northern pool. Maybe it covers more square meters per hour. That would be akin to asking if you get wetter by walking or running in the rain.

Mintaka

Well, I’d venture to say that the majority of meteorites that enter our atmosphere do so during meteorite showers. And those occur when the earth moves through clouds of debris that lie, or have come into, it’s path… Hence i’d say that the most meteorites would come at earth from a specific direction… ie, the direction that the earth is traveling along it’s orbit.

Secondly, I’m reminded that the earth is tilted. Because of that, the two pools will deviate “up” and “down” as the earth rotates, with regards to the imaginary path the earth is taking. But, because of the shape of the earth, the “higher” or “lower” a pool is, the less surface area it presents to the debris sitting out there in the earth’s path. With the optimum spot for presenting the most surface area being dead center (thus, when it’s turned such that the pool is facing the incoming debris head on, ie, the pool is on the same plane as the earth’s imaginary orbital line.). This is where it gets tricky. Sometimes the tropic of cancer pool gets rotated such that it is closer to this optimal spot than the equatorial pool. And sometimes it’s the other way around. You’d have to figure out what the ratio is… Note that this position also depends on the earth’s position in it’s orbit, because the orientation of the earth’s tilt with regards to the incoming debris changes hear round… Intuitively 1/2 the year one pool will get more exposure, and 1/2 the other. Thus I have a feeling that it cancels out…

Note that this also holds for the amount of atmosphere that debris must travel through. Sometimes one pool has the upper hand, sometimes the other…

Nope, I’m lost. If there’s a difference it’ll take quite a bit of math to arrive at the answer. There’s lots of motion going on here…

If anything, I’m going with, too close to call, probably the same.

Most meteorites found on earth was picked up in Antarctica. It is easier to spot them against the white snow and ice but that also covers a lot. So I think the futher north or south you go the better you chance.

Mintaka, it seems like you’re going to raise more questions than answers on this Topic
(from my side in anyway) ;D

Ok, so here I’m going:

According to geology.com:

“Each day Earth is believed to gain over 1000 tons of mass from the infall of tiny meteorites. Most of these meteorites are the size of a dust particle or sand grain. Rarely a meteoroid large enough to be witnessed falls all the way to Earth. Several hundred meteorites larger than marbles are thought to reach Earth’s surface each year.”

I would say the two pools would fill up at the same rate, even though earth is tillted, debris don’t enter Earths atmosphere only at a certain point. If debris were to fall more on certain areas than others, wouldn’t earth be oval shaped after 4.5 billion years of debris? 4.5 billion x 365 x 1000 tons :o

According to josleys.com We know also that it is not a perfect sphere: the diameter from pole to pole is shorter than the diameter at the equator. The difference is small: the equatorial diameter is about 12,700 kilometers, and the pole to pole diameter is only about 40 km shorter.

After 4.5 million years, a 40km difference OK OK maybe someone are prepared to do some calculations for us? :
But what if …earth were shaped like this 4.5 bil years ago, only with a smaller diameter, and debris
caused it to grow to 12, 700 kms?: Which would mean your pool levels would uuuuhhhhmmm darn ???

;D

Each year the ecliptic (the plane in which the earth orbits the sun) will intersect the equator twice, while each tropic will be brushed by the ecliptic only once. So going with your argument, the equator should spend more time at the “optimal” position, i.e. right in the midde at the front of spaceship earth as it hurtles into space debris. Interesting idea.

The slight bulge at the equator is usually blamed on the centrifugal force due to the earth’s rotation.

Oh-oh! Hope you didn’t end up anywhere nasty.

If debris were to fall more on certain areas than others, wouldn't earth be oval shaped after 4.5 billion years of debris? 4.5 billion x 365 x 1000 tons

Probably not. Gravity will soon suck everything back into a sphere. Its interesting to look at pics of the gazillion moons of the gas giants, and the larger asteroids. It seems that the bigger they are, the closer they become to ball shape. The Earth, I would imagine, is well within the mass range that will render it perfectly spherical should it stop spinning.

Mintaka

On the rain bit - does an object moving in the rain get wetter if it is not acting as a scoop? If you had two perfectly flat horizontal rectangles of glass out in the rain, one still and one moving, does the moving one gather more moss, as it were? ???

On the thicker bit, there is more water vapour closer to the equator, to cool the meteor, giving it a better chance of surviving to the pool LOL! But again, the air is warmer at the equator. You seem to be right about the thickness:

The troposphere begins at the surface and extends to between 7 km (23,000 ft) at the poles and 17 km (56,000 ft) at the equator, with some variation due to weather. The troposphere is mostly heated by transfer of energy from the surface, so on average the lowest part of the troposphere is warmest and temperature decreases with altitude. ... The troposphere contains roughly 80% of the mass of the atmosphere.
http://en.wikipedia.org/wiki/Earth's_atmosphere

That said, it does seem that there are loads in the polar regions, so my original guess is looking good!

Considering this is a logic problem, would I be correct in assuming we are not allowed to use anything other than what we can directly deduce from the question?

Considering this is a logic problem, would I be correct in assuming we are not allowed to use anything other than what we can directly deduce from the question?

Peter, you are of course correct - logic problems strictly speaking should contain all the info required, and the answer is known (at least by someone else). If there is a definite answer to this, I don’t know what it is. Perhaps calling it a logic problem, was incorrect. Maybe its more of a speculative exercise, and some information outside that given would be useful, if not essential. Thanks for pointing that out! :-[

Mintaka

No worries, I’m just considering how to think about the problem.

How did you come up with it?

OK, here’s how I imagine it:

You’re in orbit around the earth and you fire off meteors, every second, at both the equator and at the Tropic of Cancer simultaneously. The meteorites don’t all land on the same spot because the earth is rotating, but are spread out evenly along the two lines of latitude. Meteorites on the equator are more widely spaced because the earth is rotating faster there. Meteorites at the Tropic of Cancer are more densely clustered and the pool will fill more quickly.

Voting without posting shouldn’t be allowed. >:(

OK, OK. Let’s assume a meteorite flux of 1 meteorite/m^2/day everywhere on the surface of the Earth (We’ve been told that the flux is equivalent, so meteor showers and so on are irrelevent).

Assume also that the dimensions of the swimming pool are 50m x 20m = 1000m^2
If the Earth were not rotating each pool would would be struck by 1000 meteorites per day.

Does the rotating Earth make a difference?

The area swept out by the equatorial pool during the course of the day is:-
Circumference of the Earth at the equator x 20m = 4,000,000m x 20m = 80,000,000m^2

The area swept out by the tropical pool during the course of the day is:-
Circumference of the Earth at the tropic x 20m = 3,668,200m x 20m = 73,364,805m^2

This would seem to indicate that the equatorial pool should be struck by more meteorites, BUT: The equatorial pool is moving faster and therefore spends proportionally less time in each ‘sector’ of its larger area. This exactly cancels out the effect of the larger area, and each pool will be struck by the same number of meteorites.

I almost forgot another assumption: a spherical Earth. As we all know on this forum, the Earth is NOT spherical; it is an oblate spheroid, fatter around the equator than the poles, and the Southern hemisphere is slightly heftier than the Northern. Sort of pear-shaped, actually. Over millions of years, this shoulld have the effect of causing the equatorial pool to gather ever so slightly more meteorites.

There you go. You just had to bring maths into it didn’t you? ??? :-\

How? I don’t see how you reach this conclusion.

This I agree with.

It’s one or the other, I don’t see how it can be both.

Imagine Cameron Diaz. She is naked save for two narrow (say, two inch wide) white belts, one about her hips and the other around her waist. Sewn into the belts are railway tracks along which run two identical carts. The carts move around the tracks at speeds that result in their completing their circumnavigations of Cameron in precisely the same amount of time.

Now imagine the seven dwarfs standing (evenly spaced) around Cameron, each armed with a tiny paintball gun. They open fire and continue their bombardment for the time taken for the carts to complete one orbit of Cameron, then they stop.

We now remove the belts from Cameron and lay them out. The belt that was around her waist is an analogue of the strip swept out by the swimming pool on the tropic, the belt that was around her hips is an analogue of the strip swept out by the equatorial pool. We would expect that–because it is longer–the hip belt will have more paintball splodges than the waist belt, but it will have the same number of splodges per unit area as the waist belt.

Quite right. So let’s turn our attention to the carts. Because they were moving for the same amount of time through an identical paintball bombardment, they will have been struck by the same number of paintballs. Their speed is not relevant. They are the analogues of the swimming pools.

Enjoyed imagining that even more.;D However, if we are to assume the dwarves are firing an equal number of paint balls at her hip and waist belts and that they never miss, even when firing at her narrow little waist, then an equal number of paint balls will hit each belt and the splodges will be more spread out on the hip belt.

Would it be an equal bombardment? By the time a volley of seven paint balls reaches her waist their impact points will be closer together.