Projectile motion fountain question

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SUMMARY

The discussion focuses on determining the equation for the shape of the water bell formed by jets from a hemispherical rose in a fountain. The correct equation is identified as y <= v²/2g - g/2v² x², where v represents the velocity of the water jets and g is the acceleration due to gravity. The conversation emphasizes the need to establish a coordinate system, ideally placing the origin at the point where the water jets emerge. Additionally, the concept of finding the envelope of the parabolic trajectories of the water jets is highlighted, particularly for angles greater than or equal to 45°.

PREREQUISITES
  • Understanding of projectile motion and parabolic trajectories
  • Familiarity with the concepts of envelopes in calculus
  • Knowledge of basic physics principles, including acceleration due to gravity (g)
  • Ability to set up and manipulate coordinate systems in mathematical problems
NEXT STEPS
  • Study the derivation of the envelope for a one-parameter curve set
  • Learn about the physics of projectile motion, focusing on trajectory equations
  • Explore the implications of different projectile angles on water jet trajectories
  • Investigate the mathematical modeling of fluid dynamics in fountains
USEFUL FOR

Students studying physics, particularly those focusing on mechanics and projectile motion, as well as educators seeking to explain the principles of fluid dynamics in practical applications like fountains.

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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
 
Last edited:

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