19-Sept-2016: Modeling the fall of an object falling with air resistance

Modeliong the fall of an object falling with air resistance

Purpose: To determine the relationship between air resistance force and speed, and to model the fall of an object including air resistance.

Part 1: Determining the relationship between aur resistance force and speed

To be able to model the of an object including resistance, we need to come up of an equation that expresses the relationship between air resistance force and speed. Air resistance force on a particular object depends on the object's speed, its shape, and the material it is moving through. We predicted an equation of F_{resistance =kv}ⁿ , where k is the shape and area of the object.

To verify the equation we predicted, we conducted an experiment.

In this experiment, we needed a 2 meter stick, coffee filters, a computer with LoggerPro, a dark background(cloth) and a tape to make the coffee filters more visible. We conducted this in building 13, the design technology building, but we first conducted a trial run to make sure we know what to do by the time we conduct the experiment. By stacking coffee filters, we created a set of objects with the same size and shape, but with different mass. We dropped the filter(s) at a height, and video capture the fall. We started with 1 filter and stacked another after each trial. We did this 5 times.

After capturing the videos, we analyzed the data. We needed to determine the terminal velocies of the coffee filter(s) for each trial. We used the linear fitting of the postion vs. time graph, because its slope would give the terminal velocity of the filter(s). We did the same procedure for the other four trial. By determining the terminal velocity of each trial, we obtained a total of 5 terminal velocities that correlated to different masses(as we increase the nuber of coffee filters).

Since we have the terminal velocities, we needed to determine the value of air risistance force. We calculated the mass of the coffee filter because air resistance force is related with mass. Also, we measured the mass 150 coffee filters to minimize the uncertainty in measurement.

Obtaining the values of terminal velocity and air resistance force, we enterd the data in LoggerPro. We graphed our data and observed its pattern to verify our air resistance force equation.

Part 2: Modeling the fall of an object including air resistance

The Goal here is to apply the mathematical model we developed in part 1 to predict the terminal velocity of varius coffee filters.

We used Excel to model the fall with air resistance. To determine the terminal velocity, we made 6 columns containing time, change in velocity, instantaneous velocity, change in position, and instantaneous position.

To get more accurate result, we set the ∆t value smaller. Also, when the acceleration was zero, we observed that we would obtain the value of terminal velocity. We then filled the columns down until we reached a row where the accelration is zero because the velocity that correlate to that acceleration would be the terminal. We did the same procedure to the other coffee filters.

Conclusion

 In this lab, we learned how to model the fall of an object falling with air resistance. We proved our predicted air resistance force equation by determining the relationship between air resistance force and speed and modeled it using Excel. As an example, we can determine the air resistance force by calculating the force and velocity of one coffee filter  F_resistance = 0.001226*v^1.688. We also determined the terminal velocity by using Excel. Some errors that could have affected this experiment, are the wind in the building and uncertainty in measurement.