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  E \propto gHM_{spike} \propto gH(\rho H^3) = g \rho H^4, where g is the acceleration due to gravity, \rho is the density of water, and “\propto” means “proportional to”.  The energy of a falling *ehem* object is E = gdM_{"object"}, where d is the drop height.  These energies should be proportional.  Seems reasonable…  So solving for H:

H=c\left(\frac{M_{"object"}}{\rho}d\right)^{\frac{1}{4}}

Here c is some constant that would need to be found experimentally.  The graph of x^{\frac{1}{4}} 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.

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3 Responses to 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 ?

  1. christara says:

    after seeing so much of my artwork get ripped by so many it makes you feel mad.
    has with much of my art work i make for personal use for people to use as desktop wallpapers. Which is its intended use.
    or on the occasions some of my art sells thru my deviant art account.
    http://christara.deviantart.com/
    obviously if others have paid for a piece of my art so they can use it elsewhere then you may understand why i get vexed over seeing it in place’s without any recognition to myself the artist.
    or if that piece was sold then the person who purchased it would be mad at me,?
    now respectively i don’t mind a piece of my work being used in a manner than what it was intended for so if you wish to keep it on your page.
    then please do so but has long has there’s link to let people link referring back to my work http://my.desktopnexus.com/christara/ or http://abstract.desktopnexus.com/wallpaper/27127/
    and a short piece describing that its copyrighted and my name attributed to it.
    just a small piece under the artwork saying
    (Artwork is by Chrstara Copyrighted © http://abstract.desktopnexus.com/wallpaper/27127/ )
    (plus may not be reproduced, copied, edited, published, or uploaded in any Site(s) including Blogs without my written permission.) will more than suffice.
    its just so others who see it on your site will know its copyrighted work and then won’t steal it from your site and upload it elsewhere. Which makes it very hard for me to keep track of my work.?
    i thank you in advance.
    thank you chrstara.

    P.s i,ve sent you an e-mail with all different colour variations.

  2. Pingback: Q: Why does E=MC2 ? | Ask a Mathematician / Ask a Physicist

  3. Trollin Hard says:

    Thank you, … for this GREAT advice. Its absolutely, one hundred percent working. I did exactly what you explained and I didnt got a single spike. Althou I was weaving a little bit fast and had to clean 2 hours after that. It was even over my eye position, but still bellow the ceiling level, so I didnt needed a ladder to clean after. I didnt even hit the water once, but Im sure, that IF I had, it wouldnt be any spike.

    Thanks again, you deserve a nobel price!

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