## Balloon Experiment in the Classroom

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Gruszka, T. 1994. A Balloon Experiment in the Classroom. The College Mathematics Journal. 25(5): 442-444.

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“The following experiment involves a balloon, a stopwatch, and a measurement device such as a meter stick. . . . The goals of the experiment include fostering student participation, solving differential equations, dealing with parameters, comparing theory with experiment, and writing a group report.

“Experiment. The experiment consists of dropping a balloon from two different heights and recording the time it takes to hit the floor. The students are directed to complete a series of steps: A) Record the mass of the skin of the balloon. Then fill the balloon with air. B) Drop the balloon (filled with air) from a height of 2 meters. Record the time it takes to hit the floor. Repeat this a few times. Then compute and record an average reading. Call this time T2. C) Drop the same balloon (filled with air) from a height of 1 meter. Record the time it takes to hit the floor. Again average a few trials. Call this time Tv. D) Make appropriate measurements on the balloon so that you will be able to determine its volume.

“Procedures. Prior to the class period in which the experiment will be per- formed, I hand out preparation material, which consists of a description of the experiment, some questions to help them consider how the data will be gathered, and a preliminary discussion in which the governing differential equation is derived. On the day of the experiment I bring in a bag full of balloons that claim to be round, some stopwatches, meter sticks, and pieces of rope. The rope is used to make measurements on the balloon filled with air, such as axes lengths or circumferences. Prior to the experiment I weigh each balloon skin in the chemistry lab. (One could weigh a whole pile of balloons and average the results, but I noticed enough variation in the weight of each skin that I now weigh each one separately.)”

“Problem. The governing differential equation used to model the motion of the balloon, taking into account the effect of buoyancy, is *M*(*d**v*/*d**t*) = *S**g**F*(*v*), where *S* denotes the mass of the balloon skin, *M* denotes the sum of *S* and the mass of the air inside the balloon, *v*(*t*) denotes the velocity of the balloon at time *t*,* g* is the acceleration due to gravity, and *F*(*v*)) represents the resistance force as a function of velocity. The origin of our coordinate system is 2 meters above the floor with the positive direction pointing down. From the following three models for *F*(*v*)) the students determine the one that best fits their data: (i) *F*(*v*) = 0 (no resistance); (ii) *F*(*v*) = *k**v*; or (iii) *F*(*v*) = *k**v*^{2}.” Note: In models (ii) and (iii) *k* is a constant of proportionality.

Students solve the differential equations analytically in each case (i) – (iii) and from the distance the balloon fell (known!) and the time it takes to strike the floor (known!) the parameters *k* and *r*, the power of velocity in *F*(*v*) = *k**v*^{r} in the differential equation, can be ascertained.

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