Modelling 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}

Equation for the fourth/last skip in an absolute sine graph function-

y = |−0.004 + sin(2.08x − 11)|,{8.3 ≤ x ≤ 9.8}

The combination of all the four skips from the previous investigation is now created into one final absolute sine function. With the addition of limiting the domain for all the absolute sine functions to only get the curve for only the skip it formed a mathematical model of stone skipping as seen in Figure 17.

The R² value of 1 is used in order to compare how well each regression model ‘fits’ indicating that if all of the data and points fits perfectly with the regression line then the data accounts for a high degree of variance. The R² value for my first skips curve line in relation to the data is 0.9942. The R² value for my second skips curve line in relation to the data is 0.9972. The third skip was 0.998 and the fourth skip was 0.9932. Upon reviewing the graph, I noticed that not all the points perfectly aligned with the curve and the reason is that the points are merely an approximation of the stone’s path which is why my R² value is not 1. However, the data found from the graphing an absolute sine function has modelled the path of the stone skipping and demonstrated by high R² value and it can be seen that the use of an absolute sine graph is an accurate representation of the model.

I also noticed that parabolas can also fit the stone’s trajectory. Quadratic parabolas are used to create a basic ‘U’ shape of each skip through the changing of its concavity and shifting the graph to fit the points. I began by deriving an equation for each skip, given by the standard or vertex form of a quadratic function f(x) = a(x − h)² + k where (h, k) is the vertex and a is the concavity. Using the points previously found in Tables 1, 2, 3, 4, it formed the basis of the parabolas shape and equations.

Equation for the first parabola-

f (x) = −0.3x² − x + 0.1, {0 ≤ x ≤ 3.6}

Equation for the second parabola-

f (x) = −0.4(x − 5)² + 1,{3.6 ≤ x ≤ 6.6}

Equation for the third parabola-

f (x) = −1.6(x − 7.4)² + 1, {6.6 ≤ x ≤ 8.3}

Equation for the fourth parabola-

f(x) = −2.1(x − 9)² + 1, {8.3 ≤ x ≤ 9.8}

Desmos did not allow me to collate all the parabolas into one graph so here are all the parabola curves separately, with limits-

From Desmos, it calculated that the R² value for the first skip was 0.9921, the second skip was
0.8936, the third skip was 0.9035 and the fourth skip was 0.8425 (see Appendix A)

From the absolute sine graphs and the parabolas, I noticed that there is a change in the distance between each skip and that this was able to be formed into a geometric sequence.

Let u₁ = first skip, u₂ = second skip, u₃ = third skip, u₄ = fourth skip

distance of u₁ = 3.6 units (As seen in Figure...

distance of u₂ = 3.0u

distance of u₃ = 1.7 u

distance of u₄ = 1.5u

The estimated common ratio found from \(\frac{3.0}{3.6}\) is \(r=\frac{5}{6}\)

Using the formula for the nth term of a geometric sequence to work out the geometric sequence of the stone skipping model-

\(u_n=u_1×r^{n-1}\)

\(u_n=3.6×(\frac{5}{6})^{n-1}\)

The sum of a finite geometric sequence is used to find when approximately the path of this stone skipping model will end and this is given by the formula-

\(S_n=\frac{u_1(r^n-1)}{r-1},r≠1\)

\(S_n=\frac{3.6(\frac{5^n}{6}-1)}{1-\frac{5}{6}}\)

This can also be a infinite geometric sequence

\(S_\infty=\frac{u_1}{1-r},|r|<1 \)

\(S_\infty=\frac{3.6}{1-\frac{5}{6}}\)

\(S_\infty=\frac{108}{5}\)

This exploration aimed to model and graph the path of a stone skipping across water using mathematical functions. There was some success as I did this through prior knowledge of trigonometric and quadratic graphs and well as newfound knowledge through research during the exploration process, (the absolute sine function).

There were some inaccuracies in the graphing, particularly within the parabola models. As seen below, the parabola did not fit the points that I had previously plotted in the table of values and when the limits were added, the equation of the parabola did not allow for the function to accurately pass through all the points, and this is also later explored through the R² value. This is due to issues in the methodology used when I estimated the coordinates which may have carried through, also the due to the parabola

This can be improved by having more specific coordinates and refining the numerical values within the equations, for example rather than 2.5, it could me 2.54. Also, using a graphing software to make more accurate estimations of the coordinates could help improve it as it is nearly impossible to capture the stone in flight.

While the parabola did not fit as well as I would have liked, the absolute sine function graphed was able to show change between the different skips and it created a very pleasant looking function and had very accurate and reliable R² values.

I started this exploration with the idea that through the graphing of absolute sine and parabola functions that the graph would decrease both within the x and y values. However as seen in the exploration, the absolute sine graph does not satisfy these conditions as it only starts getting smaller in the x-axis. The y-value is fixed on 1 throughout as this was the only way that I could make sure that all the graphs with the skips had a repeated and accurate curve. If I did change the y-values, the sine waves would not be consistent and repeated.

Despite the first skip being an outlier and having a very reliable and high R² value of 0.9921 which suggests that it is a better fit for the regression model, the second, third and fourth skip do not have high enough values to be a good fit for the data of the parabolas. Therefore, the parabola is not a reliable and accurate enough model to fully predict and understand the path of the stone and is also not a good representation of the graph. Additionally, this large gap between the first skip’s R² value and the other skips R² values, it shows that there are some inaccuracies in my data.

With the parabolas, k is also equal to 1 because the y-value of the value is always 1 Even though, the path of the stone skipping does follow a geometric series, as seen from my calculations above, the common ratio was different between every term and there was not a clear and set number. Therefore, this made the geometric series inaccurate.

Throughout the exploration and evaluation... As further investigation, the expansion of the Maclaurin series of sin(x) or the Taylor series is an extension to this exploration that I would have liked to explore as investigating other functions and series would be interesting to see if they would fit more accurately.

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Babbs, Charles F. (2017). Theoretical Limits of Stone Skipping. Weldon School of Biomedical Engineering Faculty Working Papers. Paper 17. Retrieved on: 20^th April 2022. Retrieved from: https://docs.lib.purdue.edu/bmewp/17/

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Wonder Why. (2020). This Is Why ... skipping stones is anything but child’s play! Retrieved on: 10.12.21 Retrieved from: https://wonderwhyca.wordpress.com/2020/03/30/this-is-why-skipping-stones-is- anything-but-childs-play/.