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There is a faucet dripping water from height $H$ onto a smooth, horizontal ceramic surface, and I need to give a upper bound of the maximum distance between the water spatter and the point directly below the faucet. The naive estimate would be to think a single water drop as a bouncy ball, which on collision changes direction while maintaining its magnitude of momentum (somehow). With this assumption the maximum distance $h=H/2$. But I don't think this assumption is good enough, because water drops will break up by impact, and the momentum might be distributed unevenly, causing the smaller parts to be faster than its original velocity.

So is there any other better model for this problem? I did a quick Google search on the topic, and it seems that there are some forensics study on blood spatter, but they don't really match this scenario. And I kind of secretly hope that the viscosity of water don't play a major part in this problem (it's water, not honey, right)

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  • $\begingroup$ Please let us know what course this is from. If it's from an advanced fluid dynamics course, the answer will be somewhat different from the answer in an introductory mechanics course. The latter case is probably just asking for a combination of conservation of momentum and calculating the total kinetic energy at the point of impact, in which case I'd say "Upper bound implies perfect elasticity" and use ballistic projectile equations. $\endgroup$ – Carl Witthoft Sep 22 '14 at 13:15
  • $\begingroup$ @CarlWitthoft Actually I need this for practical reasons. But for discussion's sake I would prefer a introductory-level answer, because I've never tackled any fluid dynamics problem more complex than calculating the terminal velocity of a sphere. If my level limits the answer to ballistic-projectile level then $H/2$ it is! $\endgroup$ – arax Sep 22 '14 at 13:30
  • $\begingroup$ Could you give a bit more context of the situation? The assumptions that you can make depend on the physical details. Also, how do you know that the momentum is conserved? $\endgroup$ – user3823992 Sep 22 '14 at 13:34
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I would recommend that you derive an empirical answer by running a few experiments. Use different colored water, and measure the resulting splatters. Use the results to derive the formula.

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