Bill Gates math problem....

Quote: ItsCalledSoccer

Thanks for your patience, I'm not being contrary, ...

Didn't think you were. This is not a trivial topic.

Quote: ItsCalledSoccer

Maybe I should sleep on this ... sometimes things marinate and I can understand them better.

I understand and experience that too.

Quote: ItsCalledSoccer

... So, from a point of view on the earth's surface .. say, looking up at the falling Bill ... it would look like he's falling straight down.

Actually, no. Bill would look as if he is floating off toward the horizon. But I do understand that this is not intuitively obvious. Since you do leave the impression that you would like to understand this (a very favorable trait, in my opinion), I will go into some detail, and the rest of the forum can just keep laughing at us geeks.

Let's consider a different problem a while. It will raise some of the same issues in a slightly less complex situation and perhaps help us reach a better understanding. Then we can go back to the Bill-Gates-on-a-stack-of-money scenario. This different problem I propose really looks at the Coriolis effect that I had been hesitant to go into (not really sure why, now.) I want to reduce the problem to two dimensions, which would be quite similar to the stack of money on the equator version of the original problem. Here goes...


Suppose that you and your girlfriend (or boyfriend, depending on your gender and orientation -- no insult intended) are riding on a carousel. You are at the outside edge, while she is straight toward the center of the ride from you at a radius one-half of yours. We assume the carousel is running at a constant rotational speed, and let's assume it is in the more common, counter-clockwise direction of carousels. Both you and she (and everything else on the carousel) are traveling at the same angular velocity in radians per second, revolutions per minute, etc. Because of the difference in radii, you are traveling twice as fast (in feet per second) as your girlfriend -- that is your tangential speed. (I may have been inconsistent in terminology previously, not sure.) Both of you would feel the tendency to be slung to the outside of the carousel, as if you were in a centrifuge, but let's assume that like most carousel riders you are able to adjust to this and just enjoy your rides around the device.

Suppose now that she were to stand still (with respect to the carousel) and you were to walk straight to her. As you approach her, your speed must be adjusted to match hers. This means that as you walk your radial path, you must experience a reduction in your tangential speed. This tangential acceleration (or deceleration in this case) associated with radial motion in a rotating system (the carousel) is known as the Coriolis effect. (Note: be sure to read my later comment on this at the very end.) If you and your girlfriend actually did this little experiment, you would likely find yourself staggering as you tried to walk toward her. You might sense a "force" pushing you over to your right as you walked. In reality, the deck of the carousel would be pushing your feet to the left to slow you down to your girlfriend's speed, and your body would tend to fall toward your right.

How are we doing to this point? Are we still together?

Now let's consider a second little experiment. Assume that each of you brought your bowling balls along with you. (Don't you usually take a bowling ball to the carousel?) You stand at your original positions and gently roll your bowling balls toward each other. Do you think they roll in straight lines toward each other and hit at the mid-point? Not a chance! Since the carousel deck can't push laterally on your rolling bowling ball the way it did on your feet, your ball will not experience the decrease in tangential speed that you did while walking. Instead, your bowling ball will maintain its tangential speed and will appear to curve to the right as it rolls closer to the center. This curve to the right is the same as your body tended to stagger as that apparent "force" pushed you to your right.

Similarly, your girlfriend's bowling ball rolling outward would not be able to increase its tangential speed to match yours. It would lag, appearing to curve toward your girlfriend's right, your left. The two bowling balls would not approach each other and collide between the two of you.

Now how are we doing? Do you see why the bowling balls would follow that curved path on the carousel deck? If not, we can work to clarify that, but if you do understand, we have taken a big step to understanding why Bill Gates would not fall straight down beside the stack of $1 bills.

Back to the carousel and the bowling balls -- here's the really interesting part. I said that the balls appear to curve to the right. In reality, they are both rolling in straight lines -- it just looks like the paths are curving when they are observed by someone riding on the carousel who has already adjusted his/her frame of reference to the rotating system. Say what? Try it this way: Suppose that when you were ready to roll your ball, you picked out some stationary point, like the ticket booth for the carousel (located some distance outside of the rotating deck), and aimed your bowling ball toward that ticket booth when it was on the far side, past your girlfriend and past the center of the carousel. At the point of release, it seems as if you are also rolling the ball directly toward your girlfriend. But as the system rotates, the ball continues to roll on a straight line toward the ticket booth. However, from the perspectives of you and your girlfriend, the ticket booth is going around the carousel clockwise, and the bowling ball is following a curved path to chase it down.

Now I really do understand that this is not something that anyone would intuitively expect. But do you see that the straight line path (in the non-rotating world) toward the ticket booth, becomes a curved path for the bowling ball from the perspective of viewers rotating on the carousel? That is not a simple concept at all, but if you begin to follow that (in the 2D problem), then we are very much well down the path of understanding the 3D problem with Bill Gates.

Perhaps I should not address the 3D problem any further at all right here and just give you a chance to "sleep on" the 2D problem. Then we can try to deal with the remaining issues there before going back to the more complex problem. OK? Let me know whether you want to wade through this even more.

And for the rest of you in the forum, why are you still reading this?


Edited the morning after: After "sleeping on" this, I realized that I had said something incorrectly. The "curve" or apparent lateral acceleration of the rolling bowling ball is the Coriolis effect, and the apparent "force" that makes the bowling ball curve and the walker stagger is known as the Coriolis force. It is not a real force -- see below. I mistakenly called the walker's tangential acceleration (deceleration to stay on the radial path) the Coriolis effect, when it is actually a forceful response to correct for the Coriolis effect.

A comment on real vs. apparent forces: Newton's first law of motion said something to the effect that a body tends to remain at rest or in a state of uniform motion until acted upon by an unbalanced force. If you tie a weight to a string and swing it overhead, you apply a real force to the string, pulling inward with a centripetal force. You sense that the weight is pulling outward with a centrifugal "force", but in reality, the centripetal force is the unbalanced force on the weight, causing it to accelerate toward the center and move in a circle. The centrifugal force is an impression, not a real force; the real force is you pulling on the string making the weight accelerate. Similarly, the Coriolis force is not a real force. The bowling ball discussed above is really traveling in a straight line and doesn't have any unbalance force acting on it, certainly not a force making it curve. The person walking toward his girlfriend tends to stagger because in trying to follow the radial path, he is trying not to walk a true straight line but to spiral inward toward the girl. The forces that the carousel deck apply to his feet toward the left are real forces decelerating him while the Coriolis "forces" that seem to push him over to the right are not real forces, just vestiges of his momentum along the tangent path.

Of course, I am posting this discourse for my own entertainment, since I do understand that no one on this forum is bothering to read this thread any longer.

Quote: Doc

And for the rest of you in the forum, why are you still reading this?

Given that Bill burns up on re-entry - does he (or what's left of him) ever reach the ground anyway? Or will Bill-dust be permanently floating around in the upper atmosphere?

Here is a link to a YouTube video illustrating the Coriolis Effect. It may make it a little easier to visualize.

http://www.youtube.com/watch?v=mcPs_OdQOYU

Look at it this way. You have a baseball, and you have a basketball, spinning at 1 revolution per second. Put an object on top of a baseball and see where it lands. Put the same object on top of a basketball and see where it lands. Clearly, the velocity of the object on top of the basketball is greater than the one on the baseball. The force of gravity is a downward force while your radial velocity is and outward force. When you are introduced to the downward motion velocity of 55 m/s, there is nothing that takes you away from the angular velocity which about 10,578 / 6,378s greater. Therefore, you would indeed fly ahead of the bills and land far in front of it.

Quote: Ayecarumba

Here is a link to a YouTube video illustrating the Coriolis Effect. It may make it a little easier to visualize.

http://www.youtube.com/watch?v=mcPs_OdQOYU

Excellent! Earlier in this thread I complained that I didn't have a good way to present graphic illustrations. Well, the youtube video is better than any still sketch I could have come up with, and if I had known that link, I could have saved a bunch a words in my previous post. (I do tend to get verbose at times.)

Thanks for this contribution, Ayecarumba.