Monthly Archives: January 2012

A Rambling about Stone-skipping and its Physics

During one of our school trips a couple of years ago, we stopped by a lake somewhere to stretch our legs. Somehow we ended up stone-skipping on the lake and naturally, this progressed into a competition to see which of us could get the stone to skip the most number of times. Also naturally, I was having the worst luck and couldn’t get a single stone to skip, not even once.

Disappointed, I gave up and tried to figure out the physics behind stone-skipping. I couldn’t, and I promised myself I’d look into this mystery once I got back home.

The rest of the trip was much more exciting however (It was a rainy day, we were in a rainforest and we… uh… thoroughly experienced the dominance of leeches on the local food chains) and in all that excitement I forgot about the whole stone-skipping episode and I didn’t remember it again until a few months ago.  When I did, I immediately turned to Ask.com Google and what I found was really interesting. So I thought of sharing some of that information with you (aren’t I just so adorably generous?).

Ordinary experience tells us that if we chuck a stone into water, it would sink. Clearly this is not the case with stone-skipping and we can sort-of intuitively tell that it has something to do with the way the stone is being thrown.

Now to the physics. Newton’s Third law tells us that every force comes with a reaction force (which has the same size as the original force but acts in the opposite direction). When you’re sitting on a chair, for example, you are pushing the chair downwards and in turn the chair is pushing you back up, allowing you to (hopefully) sit comfortably without falling down. If you drop a stone into water, similarly the stone pushes down on the water and the water pushes the stone back up (or in other words, gives a “lift” to the stone). However, there is another force—gravity, that’s trying to pull the stone right down. If the lift trying to push the stone up is not as strong as the force of gravity pulling the stone down, the stone—you guessed it—falls down.

Things are slightly different in stone-skipping, and I’ll try to explain why.

A French physicist named Lydéric Bocquet has published a mathematical description of stone-skipping1 (amongst other serious publications, he has also published a paper on cooking potato wedges and soon another one of his papers on ironing and the anatomy of wrinkles will be appearing in Soft Matter—needless to say I like this guy!). He calculates that the lift experienced by the stone is proportional to the square of the stone’s velocity. In other words, the faster the stone travels, the greater the lift. By playing around with maths, he came up with an expression that gave a critical velocity (i.e. a “speed limit”) for the stone. If the stone travelled at a velocity lower than this critical velocity, it would sink. In fact, the physics here is similar to that experienced by a water-skater. If a skater is travelling very slowly, he or she obviously can’t skate on the water and would sink.

Bocquet also realized that each time the stone hit the water surface, it experienced friction and this reduced the stone’s kinetic energy. This led him to calculate another critical velocity for the stone. For a typical stone, it turns out that this second critical velocity (needed to ensure that the stone has enough kinetic energy to skip) is greater than the critical velocity necessary for the stone to get “lifted”.

The stone also has to hit the water surface at an angle. The lift is much less for a stone that lands “flat” on the water and it would almost certainly sink. Boquet experimented extensively on the angle with some other scientists—and published his results in 2004. It turns out that an angle of about 20o gives the most number of skips regardless of the stone’s velocity and other such conditions2. He also found that there is no skipping if the angle between the stone and the water is greater than 45o 3.

It is also very important that the stone is spinning. Spinning gives the stone some angular momentum, and thanks to conservation of angular momentum, the stone’s angle to the water does not change by a great deal as the stone skips (Conservation of angular momentum is also the reason why you feel much more stable when you’re riding a bicycle at a higher speed). If the stone was not spinning, it would topple as soon as it hit the water and then sink right down. So much for skipping.

So what should you do to get a good skip of a stone? Get a flat, rounded stone. Throw it fast, at an angle (preferably 20o) to the water, and take care to make it spin (without wobbling!). If you have better luck than me (you most likely will, trust me,) you’ll get the stone to do splendid skips on the water. You might even break the world record.

This is the point where you realize that stone-skipping is actually a pretty established hobby.  Stone-skipping enthusiasts even organize competitions (check out http://stoneskipping.com/ for details on the 43rd annual Stone-Skipping Tournament). The world record right now is an amazing 51 skips. Yes. Fifty-one (as opposed to my all-time high of zero). The person who set the record is a guy named Russ Byars, and his website even gives you tips on how to get the perfect skip. The man is a professional. Watch this footage of Russ Byars setting the world record, and see if you can actually count the 51 skips (and have fun doing that! :P). In any case you just have to admit there’s something awesome about the way the stone just keeps skipping…

Alright. I’ve described the stone-skipping process rather simply. The actual situation can be more complex and I don’t want to get into all that. If you’re interested I’ll add these references, you can follow up and read them (and I highly encourage you to!).

References

  1. Bocquet, L., 2002. The physics of stone skipping. arXiv:physics/0210015v1. [This paper has the mathematical description, if you’re interested].
  2. Plus Magazine, 2002. In skimming, spin’s the thing. [online] Available at: <http://plus.maths.org/content/os/issue22/news/skimming/index> [Accessed 31 January 2012]. [This gives a very nice description for an average reader, outlining the simplest aspects of maths as well].
  3. Clanet, C., Hersen, F. and Bocquet, L., 2004. Secrets of successful stone-skipping. Nature, 427 (29), p.29. [Got some nice close-up shots of a stone’s “skip” in progress].

Some other articles with “simpler” explanations that might interest you:

This next one is actually a more recent study done by scientists at University of British Columbia and University of Cambridge. They refine Bocquet’s model by adding in more parameters, especially how the shape of water in changes as the stone collides with water. I didn’t read much of it, but it looks pretty interesting:

…and that’s that. I shall see you with my next blog post. Whenever that is. I know this post is rather vague… I’ll try my best to read up and do a better job next time! ;). Adios! 😀

ETA: A slow-motion video of a stone skipping. Can’t believe I missed this! 😀