Projectile motion fountain question

AI Thread Summary
The discussion revolves around determining the equation for the shape of the water bell formed by jets from a hemispherical rose fountain. The proposed solution is y <= v²/2g - g/2v² x², but there is uncertainty about the coordinate system's origin. It is suggested to place the origin at the point where water squirts out, simplifying the analysis. The water jets follow a parabolic trajectory, and the goal is to find the envelope curve that all trajectories fall below. Understanding how to derive the envelope for a one-parameter curve set is essential for solving this problem.
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Homework Statement


A fountain consists of s small hemispherical rose which lies on the surface of the water in a basin. The rose has many evenly distributed small holes in it, through which water spurts out at the same speed in all directions. What is the equation representing the shape of the water bell formed by the jets?

Homework Equations


The Attempt at a Solution


The answer for this question is y <= v2/2g - g/2v2 x2. I have no idea how should I start? At first I thought of setting up a coordinate system. But I don't know where I should set up the coordinate? Is it at the center of the hemisphere? Or is it at the point where water squirts out?
 
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You can assume that all the water jets come out at the same spot, and put the origin there. And find the envelope. Nice problem!

ehild
 
In all water jets, the water drops follow a parabolic trajectory, determined by the projectile angle. You need to find the envelope, a curve in the xy plane, so that all trajectories fall below it. In the x, y plane, the given function is tangent to all trajectories for projectile angles greater or equal to 45°.

Have you learned how to get the envelope for a one-parameter curve set?

ehild
 
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