The motion of a stone skimming over a water surface is considered. A simplified description of the collisional process of the stone with water is proposed. The maximum number of bounces is estimated by considering both the slowing down of the stone and its angular stability. The conditions for a successful throw are discussed.

The actual world record appears to be 38 rebounds (by J. Coleman-McGhee). See, for example, 〈〉 for more information on stone skipping competitions.
Some pictures of the bouncing process of a circular stone on water and sand can be found in
C. L.
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The Amateur Scientist
Sci. Am.
H. R.
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How things work: What can a dimple do for skipping stones?
Phys. Teach.
D. J. Tritton, Physical Fluid Dynamics, 2nd ed. (Oxford University Press, Oxford, 1988), pp. 97–105.
L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Pergamon, New York, 1959), pp. 168–175.
Note that the nontrivial point is to assume that Cl does not vanish and reaches a finite value in the small θ and β limit. We may invoke the finite aspect ratio (thickness over lateral size) of the object. For example, if the stone is an ellipsoid of revolution with thickness h and radius a, with h≪a, we expect Cl∼h/a (Ref. 5). However the proportionality constant is expected to be sufficiently large so that the lift effect is non-negligible. This property is exemplified by water skiing. In this case, the lift force is sufficiently large to sustain the weight of a skier on small boards, while both tilt and incidence angles are close to zero.
It is amusing to note that the laws of friction for the stone are similar to those of solid friction. We have indeed Fx=μMg, with μ=C̃/C, independent of the velocity and surface of the stone. Of course, the same result holds for water skiing, which is not obvious.
H. Goldstein, Classical Mechanics, 2nd ed. (Addison-Wesley, New York, 1980), pp. 203–213.
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