Doc
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November 15th, 2010 at 2:30:58 PM permalink
Quote: Nareed

I assumed it would be built on the Equator, because no other location makes sense for such a structure due to the reasons you mentioned.

Ooops! It just occurred to me -- would building the tower at one of the poles (almost) eliminate the problem of not dropping vertically? If Bill were wearing his re-entry suit, perhaps his descent path would just spiral around the tower.
ItsCalledSoccer
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November 15th, 2010 at 2:47:32 PM permalink
I think that his downward drop would not be affected no matter where he was unless he had some sort of tangential velocity. If he dropped straight down, the earth would turn under him but he would still be affected by the downward pull of gravity ... so he, the stack, and the earth would turn at the same rate, right? You don't have to be connected to the earth to turn with it.

In any event, there would be some altitude where this would be the case. If you base-jump off of a tall building (altitude 1,000 ft) or out of an airplane (4,000 feet), this is certainly the case - the only "side-to-side" is provided by some initial (provided by velocity of the plane or outward push away from the building) or in-flight change (eg, parachute slows it) in tangential velocity.

In this case (assuming the stack is rigid), the top of the stack from which he jumps would be moving at some tangential speed, and when he's on the top, he's moving at the same tangential speed. When he jumps off, he's still moving at that same tangential speed. But as he proceeds down the stack, the bills that make it up are going at a slower and slower tangential speed (but the same angular velocity).

If I recall kinematics-in-a-vacuum, then the "slip" he experiences when he's up high - the earth spinning underneath him and not taking him along with it, before he reaches the "base-jump level" or "skydiving level" or whatever altitude it is before this "slip" becomes infinitessimal - is exactly made up by the difference in his original (and continuing) tangential velocity and the decreasing tangential velocity of the stack (reference point) as he descends.

I don't know the math for this, but I seem to recall some basic classroom experiments that confirmed this, like dropping a stuffed bear from a tall standard, at the same time shooting a marble originally aimed at the bear, and by the time the marble had traveled the horizontal distance to the bear-drop, it would ALWAYS hit the bear, no matter the original velocity of the marble (assuming it was enough to reach the bear in the first place).
Wizard
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November 15th, 2010 at 3:09:41 PM permalink
Quote: ItsCalledSoccer

You don't have to be connected to the earth to turn with it.



Mostly true. However, it is my understanding that very long-range rifle shots are affected by the movement of the earth under the path of the bullet, and will result in being slightly off target if not adjusted for properly. I'm not sure why, but here is a link about it.
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Doc
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November 15th, 2010 at 3:35:06 PM permalink
Quote: Wizard

...it is my understanding that very long-range rifle shots are affected by the movement of the earth under the path of the bullet, and will result in being slightly off target if not adjusted for properly. I'm not sure why, but here is a link about it.

My initial reaction is skepticism to the amount adjustment that article suggests is necessary for Coriolis effect on a rifle bullet. I would like to see some more details on those calculations. On the other hand, as I suggested above, there is indeed a significant adjustment necessary for a missile traveling a few thousand miles.
Wizard
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November 15th, 2010 at 3:37:06 PM permalink
Quote: Doc

On the other hand, as I suggested above, there is indeed a significant adjustment necessary for a missile traveling a few thousand miles.



Wouldn't the same principle apply to the bullet?
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ItsCalledSoccer
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November 15th, 2010 at 3:42:30 PM permalink
Quote: Wizard

Mostly true. However, it is my understanding that very long-range rifle shots are affected by the movement of the earth under the path of the bullet, and will result in being slightly off target if not adjusted for properly. I'm not sure why, but here is a link about it.



I think that's right but in that case, there's a great deal of tangential velocity ... which, by the way, is what keeps satellites in orbit. I guess I was assuming that tangential velocity was 0, at least that component that's not attributable to the ever-present angular velocity * radial length. After all, in our daily life, we're moving at one rotation per day (not counting the speed from the orbit around the sun), and it doesn't seem like we're moving at all!
Doc
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November 15th, 2010 at 4:10:29 PM permalink
Quote: ItsCalledSoccer

I don't know the math for this, but I seem to recall some basic classroom experiments that confirmed this, like dropping a stuffed bear from a tall standard, at the same time shooting a marble originally aimed at the bear, and by the time the marble had traveled the horizontal distance to the bear-drop, it would ALWAYS hit the bear, no matter the original velocity of the marble (assuming it was enough to reach the bear in the first place).

Your stuffed bear experiment does generally work that way, but it is very different geometry for the problem with the tower. If there were no rotation at all, when Bill stepped off the tower, he will just drop straight down, landing next to the base (ignoring the burning up on re-entry issue). With the rotation, things are different.

Suppose the base of the tower was at 40 deg North latitude. That point on the surface of the earth is rotating in a circle north of the equator. The top of the tower would also be traveling in a circle, but it would not be in the same plane as the circle of the base. The peak would be traveling in a circle even farther north. In fact, my quick calculations indicate the plane of that circle would be almost 1,000 miles above the north pole. Suppose for a moment that the velocity at the peak were such that when Bill stepped off he would maintain orbit at that altitude. He would not stay next to the tower platform. Instead he would begin to orbit around the center of the earth. Since the velocity would not really be adequate to maintain orbit (I don't think), he would fall, but not next to the tower.
Doc
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November 15th, 2010 at 4:12:28 PM permalink
Quote: Wizard

Wouldn't the same principle apply to the bullet?

The same principle applies, but it's a matter of whether the effect is significant over small distances. It is very much like the comments from your father that you noted in your blog.
Wizard
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November 15th, 2010 at 4:41:01 PM permalink
Quote: Doc

The same principle applies, but it's a matter of whether the effect is significant over small distances. It is very much like the comments from your father that you noted in your blog.



I've read different opinions. Some say that over a very long shot, like 1000 yards, it is enough to think about it if accuracy is essential, like a good head shot. I'd be interested to know how many millimeters a 1000-yard shot would be off due to the coriolis effect. It would likely depend on the latitude.

I did ask my dad about this, but have not heard back. Coincidentally, I have bothered him twice about the coriolis effect lately. For those who care, here is the blog entry Doc referred to: My father's report from the equator.
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Doc
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November 15th, 2010 at 4:58:19 PM permalink
Quote: Wizard

I've read different opinions. Some say that over a very long shot, like 1000 yards, it is enough to think about it if accuracy is essential, like a good head shot. I'd be interested to know how many millimeters a 1000-yard shot would be off due to the coriolis effect. It would likely depend on the latitude. ...

I agree. That's why I said I would be interested in seeing more details on the calculations.

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