Modeling Free-Fall Flight and Terminal Velocity of Coffee Filters

Four-Filters Falling Movie

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Activity: Modeling Free-Fall Flight and Terminal Velocity of Coffee Filters.

Objective: To help students understand the physics and mathematics of free-fall motion with a drag force due to air resistance.

For this exploration, we selected a coffee filter for our free fall flight. A coffee filter is light enough to reach a terminal velocity quickly and a number of filters can be stacked together to change the mass of the falling object. Using a ladder, we dropped the coffee filters from a height of about eight feet. We also placed a yardstick on the background wall so that we could calibrate the measurements we would make from the frames of the video. We made videos of the free fall flight of a stack of 1, 2, 3, and 4 coffee filters. We then imported the frames of these videos into TEMATH and measured the position of the coffee filter in each frame of its flight. The differential equation that models free fall flight with air drag is

m dv/dt = m g - 1/2 D p A v^2

where m is the mass, p is the density of air, A is the projected cross-sectional area of the falling object, D is the drag coefficient, g is the acceleration due to gravity, v is the velocity, and t is time. After being unsuccessful in fitting this model to the measured data, it was determined that the usually neglected upward buoyant force must be included in the model. The differential equation with the buoyant force is

m dv/dt = m g - 1/2 D p A v^2 - p g V

where V is the volume of the object. Example models and measured data for the distance fallen by a stack of two and four filters are given in the Work window.

Click the buttons to see the fit

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