By Jacob Paul Duncan

Mathematics, Winona State University, Winona MN USA

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Most projectile motion and free fall models are based on the assumption that gravity is the only force acting on the object. Here we develop, solve, and analyze a second order nonhomogeneous differential equation model for free fall which incorporates air resistance. Students will solve the model using two different methods -- reduction of order and separation of variables, and method of undetermined coefficients. Using the solution, students derive an expression for the terminal velocity of the object as well as a prediction of the maximum height of the object (assuming is is fired directly upwards). The model is then parameterized for a Nerf dart by an experiment performed by students. The terminal velocity and muzzle velocity of the dart is measured using video frames of the dart's motion. Finally, the model is validated by an experiment wherein students fire their darts upward and measure the assent time for comparisons with their predictions.

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Researchers should cite this work as follows:

  • Jacob Paul Duncan (2022), "4-039-S-FallingDarts," https://simiode.org/resources/9001.

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