Q: Is there a formula for how much water will splash, most importantly how high, and in what direction from the toilet bowl when you *ehem* take a dump in it ?

Physicist: If it weren’t for imponderables like this, we’d have finished science years ago.  During an “impact event” water generally moves outward to the sides.  What you really need to worry about is the dreaded “water spike”.

Ejecta, spike, ejecta, and spike.  (The artwork in the upper-left is by Chrstara, Copyrighted ©  http://abstract.desktopnexus.com/wallpaper/27127/. These pictures may not be reproduced, copied, edited, published, or uploaded to any Site(s) including Blogs without his written permission.)

The physics behind water spikes is remarkably complicated and only recently has their formation been accurately described and simulated.  So, like any physicist presented with an insurmountable problem, I’ll make some unreasonable assumptions and cheat (an experimentalist would then drink and make prank calls).

One of the classic cheats is making a list of everything you think your equation should depend on, and then balance the units.  Based only on the vague hope that water spikes scale (same shape regardless of size), the energy E of a spike that rises to height H should be  , where g is the acceleration due to gravity, is the density of water, and “” means “proportional to”.  The energy of a falling *ehem* object is , where d is the drop height.  These energies should be proportional.  Seems reasonable…  So solving for H:

Here c is some constant that would need to be found experimentally.  The graph of increases sharply from zero, and then sorta levels off.  So don’t expect to have to much influence on the height of the spike given that this already shot-in-the-dark equation is not strongly influenced by small changes in the variables away from zero.

Your best bet is to avoid generating the spike in the first place.  Water spikes are the result of a symmetric air-cavity collapse just below the surface.  If the cavity isn’t symmetric, you shouldn’t get a spike.  So as you make your Deposit, make sure to wave your butt around.  Please let us know how it works out.

Here’s another example of frequently gross marriage between super-computers and fluid dynamics.