# When an object plunges into water, what is the maximum height a droplet from splash can reach?

I cleaned up the kitty litter just some minutes ago.

When I threw the products of our 7kg cat into the toilet bowl, the broth splashed and one drop went straight into my eye.

yuk.

Here come the numbers:

• Falling height was ~ 30 cm
• My face was ~ 90 cm above the water surface
• I was not exactly perpendicular above the "entry point", perhaps 20 to 40 cm off center.
• cat poo was ~ 3 cm thick and between 3 and 8 cm long, rounded.
• poo surface was rather smooth. He wasn't outside to chew on grass, because it's too hot. He ate dry food in the morning and raw chicken with bones in the evening.

I didn't try to put the poo onto the kitchen scale for gaining more data in front of my girlfriend fearing to risk the eyesight on the other eyeball.

So for now: Is there or are there some indisputable distances lower than geostationary orbit where I'm safe from toilet broth when disposing off cat poo?

Update

@userLTK gave some good constraints for narrowing the problem. While pondering I came up with some subproblems which pave the way to an exhaustive answer.

To narrow the percentage of kinetic energy which can be "reused" for the splash acceleration, it may be interesting, what an optimal poo form may be. Perfect sphere, sausage like thing or something completely irregular with perhaps considerable rought surface?

This may be rather difficult to answer. So I thought about the other end of the problem: How high can a droplet rise on its own? AFAIK the maximum size of a droplet is determined by the friction in air and the resulting deformation. A raindrop falling with up to 10 m/s cannot exceed 9 mm in diameter without beeing disrupted by air pressure differences. Is the maximum air speed known for different drop diameters? If so, the initial speed of a single droplet splash is limited. Then we can calculate the maximum elevation with or without air drag.

This reduction of the problem lacks consistency if initially splash consists of bigger "conglomerates" of water. Those amorphous water masses may reach higher speeds until they break up. I suppose, this maximum speed subproblem is probably the strongest constraint to the height of the splash.

• I don't want to take away from your question, but Kitty Poo, according to some, shouldn't be flushed: superterry.com/can-cat-litter-flushed-toilet and moderncat.net/2008/11/13/toilet-training-your-cat - Jul 19, 2015 at 3:15
• Hum, I'll consider that. Following this suggestions the task would be altered substantially. Jul 19, 2015 at 13:34
• Reading the comments, there's different points of view, like indoor cats it's more likely safer to flush than with cats that go outside. Jul 19, 2015 at 13:47
• We go out with the cat on a leash. So it is very unlikely for him to get infected with toxoplasma by feeding on mice. But nevertheless he may be carrying toxoplasma. On the other hand our municipality started to introduce disinfection stages to the wastewater treatment and the wastewater travels ~ 2500 km till it reaches the black sea. Jul 19, 2015 at 13:57

On the water-splash, here's a video that I think explains what happened. https://www.youtube.com/watch?v=2UHS883_P60 (the 3 balls, not the double bounce on the trampoline).

The Newton law of conservation of energy says that energy is concerved, so a share of the energy of the drop transfers to the energy of the splash.

Figure you drop it from 1/2 meter and the mass is 0.2 KG (about 7 OZ). Do a little math and 1/2 * 9.8 * .2 = .98 Joules, or about 1 joule of energy.

How high a splash can 1 joule make in still water, well, it depends. The energy is converted into 3 different types of energy, wave energy, which is likely the biggest share, splash energy (#2), which depends on a few factors like the shape of the object, depth of the water, if there are already waves present, etc, and the 3rd and likely lowest of the 3, heat energy. Here's where the YouTube video comes in. If it splashes just right, you can get much more velocity out than you put in, into a much smaller mass.

There's no good calculation that I know of to give you maximum height a water drop can reach in this scenario. It depends on how well the energy transfers into a drop and what size drop.

As you know from experience, this kind of splash-back is rare, but dropping from a lower height or dropping one at a time vs 2 at the same time should reduce the likelihood of a high splash - that's probobly super obvious.

A curious and loosely related field of study is wave size from large meteor impacts or landslides into the ocean. A rather fun one to read about is the Canary Islands waring, that one research team made and others think they exaggerated, but it found it's way onto television and there were warnings that a Canary Islands landslide could create 50 foot waves across the entire eastern seaboard". Probably not accurate, I think, but the warning made some impressions all the same. More on that here: http://news.bbc.co.uk/2/hi/science/nature/3963563.stm

• This answer is a good start. Taking the energy into account explains why the drop was able to reach a higher altitude than that where the poo was dropped from. The hint to the momentum is interesting. But exchange of momentum is AFAIK valid for fully elastic collision. Can this model (as described in the video) be transferred to the complex exchange in a fluid? Jul 19, 2015 at 21:40
• What little I know, the modeling of water is crazy complicated. Like, NASA complicated. Here's a loosely related article on the difficulty Pixar had generating flowing water (scroll down to water generation). cnet.com/news/… Another way to ask your question is, if you drop identical rocks from identical heights into still water, can you generate a consistent splash, and unfortunately I'm not smart enough to answer that with certainty but from my observation, (and this is going to sound crazy obvious) splashes vary. Jul 20, 2015 at 22:38