Cowa-Bungee

Mathematical Modeling

Richard Hitt
University of South Alabama
16 March 2000

Introduction

Cowa-bungee was a new event in the 2000 High School Science Olympiad.  A description of it, taken from the official rules of the olympiad, follows.

COWA-BUNGEE

Description

Each team will provide one elastic cord with which they will conduct two separate drops (Drop 1 & Drop 2) with two separate masses.  Each team will attempt to get each mass as close as possible to, but without touching, a landing surface (or plane).  Devices will be impounded prior to the posting of both masses and both heights (which will be the same for all teams)  and no physical alterations may be made to the devices cord once it has been impounded (with the exception of marking proper drop locations on the cord before the drops).

A Team up to: 2                               Approximate Time:  15 minutes

The Competition

The masses:  Both drop masses, Mass 1 and Mass 2, will be between 25-500 grams. The masses will consist of different amounts of sand which may be placed into a 500-591 ml soda bottle or other suitable container.  The mass values and container length will be posted soon after impound.

The drop heights:  Both drop heights, Height 1 & Height 2, will be between 3-7 meters (except at Nationals where the drop height may reach 10 meters).  The heights may be the same or different.  The exact height from which the drop must occur will be verified by at least two separate measurements by the judges.  Teams may bring and use their own measuring device, but they must measure during the time given for their two drops.  The drop height values will be posted soon after impound.

The elastic cord:  Students will provide only one elastic cord to be used for both of their drops.  The cord used may consist of more than one material and more than one strand.  The cord will be considered elastic if it passes the following test.  The bottom 2 meters of the cord must stretch to at least 2.5 meters with a single 500 gram mass is attached to this bottom section while being suspended vertically and return approximately to its original length after the mass is removed.  As long as the cord passes this test, non-elastic components and materials may be used in various locations within the cord.  Any team that fails this elasticity test will be allowed to compete, but will automatically rank behind all teams which passed.  The spirit of the rules is that no team attempts to build a self-limiting-break mechanism such as a separate, parallel, non-elastic strand which breaks the fall of the mass with little to no rebound. Any team that blatantly breaks the spirit of the rules, as deemed by the event coordinator will not be allowed to compete and will receive no points.  Every team's cord must terminate its bottom end with a closed metal ring.  The diameter of the ring opening must be approximately ½ to 1 inch (e.g., a key ring).  Event supervisors will supply an attachment mechanism (hook, clasp, carribeener, quick-link, etc.) that will connect the team's bottom cord ring to each mass.  The length or height of this attachment will also be posted soon after impound at nationals.

The Drop:  Event supervisors will provide a mechanism to attach the top end of a team's cord to the specified drop height.  This will consist of some sort of platform and hook, clamp or ring.  If  teams require a more complicated top-anchoring system, they must bring it with them and impound it along with their cord.  In ALL CASES, the top of the cord must be anchored to the exact drop height in a RIGID way.  Event supervisors will also need to provide an accurate system for determining how close a team's device came to the landing plane, and whether or not it touched.  Some successful methods for determining the closeness of a drop to the landing plane include:  high-speed or digital video cameras, multiple spotters, & sonic ranger devices.  Possible methods for determining whether the device touched or broke the landing plane include:  a carbon paper drop area, very fine powder landing area, or sonic ranger devices again.  Teams will be given a total of 5 minutes to prepare their device in the holding area, followed by 5 minutes to complete both drops.

Scoring

The team with the lowest total score for the two drops will be the winner.  If there is a tie, the team with the single best drop overall (closest to the landing plane on either drop) will win.  Teams with one drop that touches the landing plane will rank below those that have no touch.  Teams with two touches will be given one point for participation.  Teams that failed the elasticity test will rank below all those that passed.

Setup

We need a few parameters for the problem:

m = mass
	L = height of the drop
	λ = length of cord used for the drop
	k = Hooke's law constant for the elastic cord

We will assume, at least for now, that the cord has negligable mass.  We will also assume that the elastic cord is 2 meters long and that non-stretchable stirng attached to the end of the 2 meter piece of elastic is used for lengths longer than 2 meters.  With this setup, k is constant.  If we use elastic for the entire length λ then k would depend on λ.  Finally, we assume that all the motion is along a vertical continuum so that we can coordinatise the problem using just a y-axis.  We take the origin to be to located at length λ below the anchor point of the cord (just where the cord becomes taught on the drop).

Calculations

At first, the forces acting on the mass is just due to gravity.  Once the cord becomes taught, however, we have a Hooke's law force acting on the mass as well.  We can use a single differential equation to model this rather that solving two different ones and splicing them together at the moment in time when the cord becomes taught.

For a mass released from rest from a height y=λ, we need to know how long the mass is in free-fall so we know when to apply the restoring force due to the elastic.  We compute this length of time first.

DSolve[{y''[t] == -g, y'[0] == 0, y[0] == λ}, 
      y[t], t]

We want the value of time that makes y[t]=0.

Solve[λ - ((g (t^2))/2) == 0, t]

Now we can set up the differential equation for the mass.

Sine the solution will be done numerically rather than algebraically, we give the parameters values to begin.  After we run through one example we can look at ways to set this up more succinctly.

We can probably get better estimates from the model if we use the stiffest elastic allowed, so lets assume our elastic stretches only 0.5 meters when a mass of 500 mg is attached to it (the least stretch allowed by the rules).  This would give a spring constant of 9.8.

g = 9.8; m =  .5; k = 9.8; λ = 3;

\(DESol = 
	
      NDSolve[
		{
          m*\(\(y'\)'\)[t]\  == 
            If[t < Sqrt[2*λ/g], \(-m\)*g, \ \(-m\)*g - \ k*y[t]], \ 
          y[0] == λ, \ \(y'\)[0] == 0}, y, {t, 0, 4}]\)

Plot[Evaluate[y[t]] /. DESol, {t, 0, 4}]

To find the minimum y-value, we use

FindMinimum[First[y[t] /. DESol], {t,  .1}]

To keep only the y-value and discard the t-value, we use

Amplitude  =  -First[%]

Then the total stretch is

TotalStretch = λ + Amplitude

Putting It All Together

Now, we can include all of this in a short Module that will generate a table of data.

Clear[Stretch]

\(Stretch[g_, k_, λ_, m_] := 
	
      Module[{}, 
		
        DESol = 
	\ \ \ \ \ 
          NDSolve[

            		\ \ \ \ \ {
              m*\(\(y'\)'\)[t]\  == 
                If[t < Sqrt[2*λ/g], \(-m\)*g, \ \(-m\)*g - \ k*y[t]], 
              \ y[0] == λ, \ \(y'\)[0] == 0}, y, {t, 0, 10}, \ 
            MaxSteps -> 10000
			]; 
		\ \ \ 
        λ - 
          First[FindMinimum[First[y[t] /. DESol], {t,  .1}
		]
	]
]\)

We can calculate the values L[m,λ] and store them in a table.

\(\(Lvalues\  = 
	
      Table[
		N[Stretch[9.8, 9.8, λ, m], 3], 

        		{m, 0.025,  .5, 0.025}, {λ, 2.9, 6.5,  .2}
	]; \)\)

TableForm[Lvalues, TableSpacing -> 1,  
      TableHeadings -> {Table[m, {m, 0.025,  .5, 0.025}], 
          Table[λ, {λ, 2.9, 6.5,  .2}]}]

Since the L values are in the body of the table, if we are given an L value and a mass value, then we can look up the L value along the row corresponding to mass and estimate the length of cord that should be used from the column label.

We can get a feel for how well linear interpolation will work to estimate intermediate values by noting that the surface is approximately linear.

ListPlot3D[Lvalues,  MeshRange -> {{0.025,  .5}, {2.9, 6.5}}]

A contour plot show this also.

ListContourPlot[Lvalues,  MeshRange -> {{0.025,  .5}, {2.9, 6.5}}]

Exercises

Exercise 1

Discuss any ways you can think of that would improve this model.

Exercise 2

Try it out!  Get a piece of elastic and test the model.