Modeling and investigating the mathematics of stone skipping

Stone skipping, a practical activity, seen as a trivial game that can be done anywhere without much thought or skill. However, this is what interests me, how can something so simple be able to become complicated enough to form perfect curves. Everything from the way the stone skims the water at a rapid speed that is not even humanely possible to see with the human eye, is something so fascinating that I wanted to investigate it more. As someone who has some prior knowledge of trigonometric functions and sequences and series, I simply want to try to find a way to understand this phenomenon by taking up a challenge to record it all down and turn it into a mathematical model using different functions. For this investigation, I asked my father to skip the stones.

The aim of this exploration is to investigate the ways in which different mathematical functions can model the path of stones skipping across water.

In theory, the distance between skips should decrease as the number of skips increase. Therefore, in the Desmos, the graphs should be curves that are getting smaller and smaller.

Due to the difficulty of capturing the stone in flight, even at a slowed down speed, the points on the graph were found from the visible ripples where the stone hits the water at x = 0 and this is seen through the arrows in Figure 2.

In order to mathematically represent the stone skipping path, I estimated and recorded the points for the trajectory of the path using Desmos. I aligned the original photo’s exact distance of 9.9m into a graph in Desmos and from there, I brainstormed and researched the potential functions that I could use to model the stone’s path.

I figured that the shape of the stone’s trajectory is a graph of a sine function, sin (x) however I also noticed that I could use the graph of an absolute sine function which has limits, providing a much better representation of the results. The function y = sin(x) is a curve or wave that oscillates between the values, -1 and 1 and in which the shape repeats itself every 2π. However, varying x, either slowly or quickly will influence the frequency of the oscillation.

The stone skipping model will mimic the curve and trajectory of the function, f(x) = |sin x|, given the assumption that the stone skipping starts at t = 0.

The Desmos absolute sine function equation-

\(y=|a+sin(bx+c)+d|\)

Where a = amplitude, b = period shift, c = phase shift, d = vertical shift

For all four skips, I began by deriving an equation for the shape of each skip by estimating the coordinates into Desmos and using the sine function formula, y = |a + sin(bx + c) + d| to create an equation, table of values, graph and r-squared, R² value.

In this investigation, I started off by working out where the stone touched the surface of the water. It was at (3.6, 0) and from there I found the maximum turning point from the middle value of 0 and 3.6. Then, based on the r-squared value, estimated the x and y-values. This created a table of values (Figure 1), a curve and an equation for this part of the function seen in Figure 3.

Equation for the first skip in an absolute sine graph function-

\(y=|-0.004-sin(0.854x-0.033)-0.004|,{0≤x≤3.6}]\)

Equation for the second skip in an absolute sine graph function-

y = |−2.49 + sin(1.06x + 2.45) + 2.51|,{3.6 ≤ x ≤ 6.6}

Equation for the third skip in an absolute sine graph function, derived from Desmos-

y = |− 0.004 + sin(1.838x + 2.696) − 0.004|,{6.6 ≤ x ≤ 8.3}