Solving Raindrop Paths on Ellipsoid with 4x^2+y^2+4z^2=16

In summary, the conversation discusses the path of raindrops on an ellipsoid surface. The raindrops are affected by gravity and will follow the path of the negative gradient of the surface. To find the specific curve, the gradient at the top of the ellipsoid can be determined and used to find the orientation of the gradient. This yields a relationship between x and y, which can then be plugged into the original equation to solve for z and determine a cross section of the ellipsoid. The process involves vector calculus and knowledge of the gradient of a surface.
  • #1
Haftred
55
0
We have an ellipsoid with the equation 4x^2 + y^2+ 4z^2 = 16, and it is raining. Gravity will make the raindrops slide down the dome as rapidly as possible. I have to describe the curves whose paths the raindrops follow. This is probably more vector calculus than physics, but i wasn't sure how to solve it, so i don't know if knowledge of physics is necessary. I found the gradient, and i know that the rain will probably flow where the gradient is largest; however, i don't know how to go on.
 
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  • #2
my guess is you need the Euler-Lagrange equation... trying to minimize the time, time=integral(ds/v)...
 
  • #3
we haven't learned that quite yet, but thanks; I have a strong feeling the gradient of the surface is involved, but I don't see how one can get the curves by taking the path of maximum descent.
 
  • #4
You can certainly answer this by finding the gradient...however, the drops will follow the path of the NEGATIVE gradient. To me, this should be a sufficient answer. To find a CURVE you would need to know the specific point from which you assume the rain drops originate on the ellipsoid (the top of the ellipsoid?) If the drops originate from the top of the ellipsoid just determine the direction of the gradient by plugging in the point. Then you can determine the line that describes the orientation of the gradient, which is a relationship between x and y. plug in y as a function of x into your original equation and solve for z. The result is z as a function of x, which represents a cross section of the ellipsoid.
 

1. What is the significance of solving raindrop paths on an ellipsoid?

Solving raindrop paths on an ellipsoid allows us to better understand the trajectory of raindrops on a curved surface, such as the Earth. This can help us predict the movement of precipitation and improve weather forecasting.

2. How is the equation 4x^2+y^2+4z^2=16 related to raindrop paths on an ellipsoid?

This equation represents an ellipsoid with a specific shape and size. By plugging in different values for x, y, and z, we can determine the path of a raindrop on this curved surface.

3. Can this method be applied to other shapes besides an ellipsoid?

Yes, this method can be applied to other curved surfaces, as long as the equation for the surface is known. However, the calculations may become more complex depending on the shape.

4. How does solving raindrop paths on an ellipsoid differ from solving on a flat surface?

Solving raindrop paths on an ellipsoid takes into account the curvature of the surface, while solving on a flat surface assumes that the surface is completely flat. This can lead to different results in terms of the path and velocity of the raindrop.

5. What are the practical applications of solving raindrop paths on an ellipsoid?

In addition to improving weather forecasting, this method can also be used in fields such as agriculture and hydrology. It can help us understand and predict the movement of water on curved surfaces, which is important for managing irrigation systems and predicting flooding events.

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