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http://img842.imageshack.us/img842/3198/bild8x.png

Solution:

We first want to determine the direction of the normal vector as Tao throws the pillar. This will serve as our initial value when we later solve the problem using the Runge-Kutta method, also known as ode45 in MATLAB.

We decide to use ode45 after we find out that the problem isn't stiff, which will reveal itself later.

We use the following coordinate system:

http://img546.imageshack.us/img546/4855/klot.jpg

Where the earth rotate around the y-axis.

So in order to determine the direction of the normal vector before Tao throws the pillar we examine the following scan.

http://img809.imageshack.us/img809/8265/bild4u.png

If we project the direction Tao's aiming the pillar in at the yz-plane we get the angle: 26.5076 degrees.

http://upload.wikimedia.org/math/b/4/f/b4f611fd056b94a7762a6e3181e7975e.png

From this scan we get how y is relative to z.

http://img810.imageshack.us/img810/3711/bild5z.png

Through the same method we have that the angle between y and z is 33.5310 degrees.

From this we can set up the normal vector as:
http://img43.imageshack.us/img43/3153/bild10ni.png
http://img88.imageshack.us/img88/6253/bild11h.png

Note that the 2nd norm of the vector is 1, just as we want (a small control check).

Now that we have the initial direction we just have to set up the equations for the forces acting on Tao during his travel. The most significant of which are the drag and the Coriolis force.

We start off by finding an expression for the Coriolis force.

http://img197.imageshack.us/img197/1741/bild12k.png

The drag force is is a bit simpler than that:

http://img715.imageshack.us/img715/6016/bild13iw.png

Keep in mind that the air density is a function of the height (basic fluid mechanics) accordingly:

http://img221.imageshack.us/img221/2263/bild14.png

Lastly we determine the drag coefficient by numerically integrating over Tao's body in the following picture.

http://img829.imageshack.us/img829/585/bild1by.png

The pillar is assumed to be out of solid marble, which has the density of 2563 kg/m^3.

Write down the data in your program and optimize for velocity.

I get the initial velocity 7100 m/s (with two digits of precession) and the following trajectory plot.

http://img864.imageshack.us/img864/7091/taotuq.png

Take note of how important the drag was.

Answer

Tao's initial velocity is 7100 m/s, or mach-21.

AWESOME!

I talked about something similar to this in the Goku-Snakeway thread.


And, excellent thinking on air density changing with elevation. I would have overlooked that.

Also, did the coriolis effect change the values of the calculation by very much? (I don't think it would affect (pun intended?) the calculation that much. But you did use 2 decimal places of precisions?)

Originally posted by dadudemon
AWESOME!

I talked about something similar to this in the Goku-Snakeway thread.
It's not really the same situation.

Originally posted by dadudemon
And, excellent thinking on air density changing with elevation. I would have overlooked that.
That's careless of you. It's going to have a severe impact on the result.

Originally posted by dadudemon
Also, did the coriolis effect change the values of the calculation by very much? (I don't think it would affect (pun intended?) the calculation that much. But you did use 2 decimal places of precisions?)
Well it made the path curved, but generally no it didn't change it much.

Originally posted by Astner
It's not really the same situation.

It's similar, not the same, ergo the words. Mine involves a constant velocity at the same altitude through a fluid. I suggest STP.


Originally posted by Astner
That's careless of you. It's going to have a severe impact on the result.

I lied: it's called a compliment. I definitely would not have over looked that. (Hence why I stated "STP" in my Snakeway suggestion. )

Learn how to take a compliment.

If you continue being an ass like this, I will no longer talk to you about these adventures. You'll post them and no one will respond to them. No one will care.


Originally posted by Astner
Well it made the path curved, but generally no it didn't change it much.

I know it didn't, that's why I brought it up. At even longer distances, it does come into play, more and more.. It all depends on how precise your professor asks you to make the final answer (2 decimal places, 3, etc.)

No one cares.

Originally posted by RE: Blaxican
No one cares.

I agree: I was the only one that cared but since he's just a jerk pretty much all the time, these days, I figure he'll stop posting/caring if no one responds to his posts.

I'll probably use the thread if I ever get into a Tao-related debate.

Just saying.

Well to start the Coriolis force isn't a real force so it has no affect on it's velocity to a observer that isn't on the ground. I'll write more once I figure out what's he's trying to do with the cross product.

This calc is ridiculous. Numerically integrate with respect to what? How can you integrate his body if you don't even have a function? You would have to approximate his body as some simple figure like a line or a cube. If you use a 3D figure most likely this will be a triple integral using some kind of Polar or spherical coordinates. The normal vector is the normal force which Tao's hands exert on the pillar. "Write down the data in your program and optimize for velocity.". That makes no sense if you have the numbers already then you would know it's velocity there is nothing to optimize. The Drag force equations changes at low and high speeds so it not very accurate. This calc is garbage plain and simple.

You seem to like Math. I shall call you Mr. Pi!

Originally posted by 3561052011
This calc is ridiculous. Numerically integrate with respect to what? How can you integrate his body if you don't even have a function? You would have to approximate his body as some simple figure like a line or a cube. If you use a 3D figure most likely this will be a triple integral using some kind of Polar or spherical coordinates. The normal vector is the normal force which Tao's hands exert on the pillar. "Write down the data in your program and optimize for velocity.". That makes no sense if you have the numbers already then you would know it's velocity there is nothing to optimize. The Drag force equations changes at low and high speeds so it not very accurate. This calc is garbage plain and simple.

He accounted for changing densities as a function of height, by the way.

And you can come up with a general idea of drag force via orthographic projection in a fluid. I think Astner accounted for that.

I believe the drag coefficient for a human body is generally .45 (I don't remember what we used)

It is a system of the pillar and Tao's body. So you must use both with their own individual contributions to the entire system. How much more did Tao's body contribute to the drag force above and beyond the pillar?


However, I agree that the coriolis effect would play very little into the calculation because the pillar and Tao were in the air for 30 minutes. But using it over such a large distance is not a bad idea if you want to be precise.

I have no idea why you decided to bump a five-month old thread. But due to the previous lack of questioning as well as your flamboyant and ignorant approach I feel that I'll have to address it.

Originally posted by 3561052011
Well to start the Coriolis force isn't a real force so it has no affect on it's velocity to a observer that isn't on the ground. I'll write more once I figure out what's he's trying to do with the cross product.
You're right in that the Coriolis force is a fictive force but what you fail to realize is that in a spherical coordinate system--which is the easiest to work in for this problem--necessitates the accounting of the Coriolis effect as you should've seen in the deduction of the formula.

To put it simple enough for you. If the earth rotates and you're traveling a global distance through a given non-earthbound trajectory you're not going to land at the same spot as if the earth didn't rotate.

Originally posted by 3561052011
This calc is ridiculous. Numerically integrate with respect to what? How can you integrate his body if you don't even have a function?
Do you even know what a numeric integration is? A numeric integration is a method to find the solution to an integral which isn't necessarily defined. In this specific problem I used MATLAB to determine the amount of pixels within the outlining drawing and then scaled it with Tao's length--as I specified--for the area in quadratic meters. Though you should be able to solve for the area with Photoshop. It's not that advanced.

Originally posted by 3561052011
You would have to approximate his body as some simple figure like a line or a cube.
Firstly, no I wouldn't have to. Secondly it's not as impressive or accurate and it's not as if it requires any great effort on an intellectual level.

Originally posted by 3561052011
If you use a 3D figure most likely this will be a triple integral using some kind of Polar or spherical coordinates.
You're confused. When I solved for Tao's and the pillar's orthogonal area I didn't use any coordinate system, and if I would've it couldn't have been a spherical one needing solving with a triple integral. I wouldn't have used polar coordinates because they're impractical to find Tao's and the pillar's area. However, if I would've used a coordinate system I'd sticked to general coordinates and in a step by step procedure creating one fitting the figure. Finally, I'm not deducing an expression in forms of variables relative to the system's trajectory, because it doesn't affect the body.

Area: A body spanning 2 dimensions.

Originally posted by 3561052011
The normal vector is the normal force which Tao's hands exert on the pillar.
No. A normal vector is a vector with the length one generally used to depict directions. In other words it has nothing to do with the force exerted on the pillar. It simply points in what direction Tao is throwing the pillar.

Originally posted by 3561052011
"Write down the data in your program and optimize for velocity.". That makes no sense if you have the numbers already then you would know it's velocity there is nothing to optimize.
I created a numeric function program based off the previous established formulas and units to find the velocity. Better formulated as: "With this angle, this distance and under these conditions what's the lowest possible velocity required to solve the problem?"

Originally posted by 3561052011
The Drag force equations changes at low and high speeds so it not very accurate.
No, there were two cases. Laminar flow and turbulent flow, both were accounted for but the latter didn't come into play until the last few hundred meters if I recall correctly and thus wouldn't have had to be accounted for.

Originally posted by 3561052011
This calc is garbage plain and simple.
I'd criticize your ability--or lack thereof--to comprehend elementary mathematical procedures.