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

[Haftred]

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.

 

[tim_lou]

my guess is you need the Euler-Lagrange equation... trying to minimize the time, time=integral(ds/v)...

 

[Haftred]

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.

 

[leright]

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.