A perfect July morning. You quietly prepare a cup of coffee and slip out of the cottage to head down to the lake. The surface is so still – the treeline on the far shore reflected in the surface like nature’s Rorschach test. That perfect surface calls out for a little physics fun, so you go in search of the perfect skipping stone: not too big; not too small; smooth and flat and round. With much experimentation over the years, you’ve got it down to an art – the perfect launch speed, the perfect launch angle, and just enough spin – the stone skitters across the surface leaving a satisfying trail of ripples in its path.
Here’s the incredible thing – there’s a heck of a lot of physics built into a successful stone skip. In the early 2000’s a group of scientists led by Lyderic Bocquet at the French National Centre for Scientific Research (CNRS) took a rigorous approach to understanding what goes into the perfect throw. You might have a feel for the ideal parameters, but this team can give you pretty precise numbers!
There are four key ingredients for a successful stone skip:
1) Speed (higher initial speed = longer path before sinking, greater # of bounces possible)
2) Spin rate (faster spin rate = greater stability, greater chance of bouncing repeatedly)
3) Angle of the stone with respect to the water surface (ideally a little bit bigger than zero, called the tilt angle, given the symbol alpha)
4) Angle of the stone’s direction of motion when it hits the surface (ideally moving mostly parallel to the water and a little bit down, called the impact angle, given the symbol beta)
It might seem a little confusing, but the two angles here are not the same thing. You can physically see the tilt angle, but it’s harder to picture the impact angle, beta. If the stone is travelling horizontally across the water, beta = 0 and if the stone is dropped straight into the water, beta = 90⁰, regardless of how the stone is physically oriented.
Getting stones to skip depends on the combination of all these variables – it’s actually pretty crazy that we can do it!
The French team conducted exhaustive experiments with their aluminium “stone” with radius of 2.5 cm, and thickness of 2.75 mm. They found that spins of about 40 rotations per second and higher resulted in greater stability for the stone to maintain its tilt angle as it bounced. They also found that a tilt of about 20 degrees was a ‘magic’ value, since it seemed to result in skipping at the lowest possible speed, as well as working well for a wide range of beta values. So maybe ‘magic’ here really means ‘more forgiving’.
The challenge with achieving lots of bounces and a long run is that the stone loses a bit of energy to the water with each interaction. The flatter the tilt angle, the less energy it transfers to the water as it bounces – much like a canoe paddle slicing through with its blade parallel to the surface rather than straight up and down in the water. So, a flatter tilt angle should go further and bounce more. The trouble with this trajectory is that successive bounces don’t go as high above the surface, so a little bit of chop in the water and the record-winning run is over.
There is a lovely paper by Charles Babbs in The Physics Teacher that derives a simplified model of the mechanics of skipping stones, but without the complications of considering spin. Naturally I had to investigate the math … here’s what happens when you change the tilt angle from 10 to 25 degrees while keeping everything else the same:
Each stone follows the same black curve to the first bounce, but then things change dramatically after that. The steeper the tilt angle, the more energy it loses with each bounce and the more steeply it bounces out again, as seen in the lab by Bocquet and crew. The flattest tilt angle travelled the furthest with the most bounces – a total of 14 complete hops (purply-pink curve) before petering out in pitty-pat (yes, that is the technical term!).
Bottom line? Like so many other things, there is tons of cool physics behind the simple art of stone skipping. Keep experimenting!
Further details? Here are the papers I’ve mentioned:
Charles Babbs paper in The Physics Teacher: https://aapt.scitation.org/doi/10.1119/1.5098910
French team papers: