Lab 4 (5th): Modeling air resistance

Title:
Modeling the fall of an object falling with air resistance

Purpose:
This lab will produce a numerical model to predict the motion of a coffee filter falling with air resistance by using video analysis to determine a relationship between air resistance and the velocity of the object.

Theory:
Air resistance is a force acting all all objects moving through a gas. When in free-fall, the air resistance on an object eventually equals the force of gravity acting on the object, resulting in terminal velocity where the object is no longer accelerating vertically. It follows then that air resistance must increase gradually as an object falls, thus relating it to the velocity of the object. Assuming all other factors such as object surface area, object shape, air density, and gravitational acceleration are constant, it is possible to produce a model relating air resistance to the object's velocity if air resistance and velocity are the only changing variables. The expected equation form is F=kv^n where n is a value relating v to F and k is a constant accounting for everything else.

Apparatus:

• Coffee filters (6)
• Long opaque black cloth approx. 6m in length with 2 tape markers 1 meter apart
• Computing device with Excel and Logger Pro installed with video capture capabilities

Procedure:

Hang the black cloth vertically from any location with enough height to allow a falling coffee filter to reach terminal velocity. Prepare to video capture in Logger Pro by orienting the computing device with camera facing the black cloth so that the motion of the coffee filter can be recorded from rest and the two tape markers are in the frame. Start the video capture once the coffee filter is ready for deployment. From the top of the black cloth, release a single coffee filter so that it follows an approximately straight path down the length of the black cloth. Stop the video capture once the coffee filter has come to rest or once it has left the video screen. Prepare 4 more separate video captures using 2, 3, 4, and 5 coffee filters stacked at once. Conduct video analysis of all 5 video captures, using the two tape markers on the cloth as 1 meter references and setting frame skip to 4 frames per data point. Using the resulting graphs, perform linear fits using approximately 5 points near the end of each graph to produce slopes representative of terminal velocity for each set of coffee filters. Plot these velocities against mass of each system and perform a power fit in Logger Pro to produce values for k and n. Using Excel, create a model with the equation F=kv^n.

Data and Graphs:

Analysis:
Our video analysis produced the first 5 graphs shown above, each with their linear fit equation shown. The values calculated in Excel included change in velocity, average velocity, acceleration, change in position, position, and net force acting on the coffee filter. The time intervals used were based on the frame rate of the video camera itself at 30Hz. Using the values for k and n produced by our experiment, net force was calculated based on values of average velocity at each time increment. A graph is shown in the first Excel data table to visualize the change in velocity based on time.

Conclusion:
This experiment produced a usable model for the motion of a free-falling coffee filter with air resistance. The exact relation as calculated by various experimental groups differed slightly, however all models related air resistance to a power of velocity.