13-March-2017: Modeling the Fall of an Object Falling with Air Resistance

Lab#4: Modeling the Fall of an Object Falling with Air Resistance
May Soe Moe
Lab Partners: Ben Chen
13-March-2017

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

Introduction/Theory: We wanted to determine the relationship between air resistance force and speed. We did not know air resistance force of an object. But we presumed that air resistance force of an object depends on that object's speed, its shape, and what it is moving through. Therefore, we modeled the fall of an object as a power law:

in which k includes the shape and area of the object. We did not know the air resistance force. So we approached this by finding out its motion, velocity and acceleration through its fall and derived the model to a line equation y=mx+b. We figured out the velocity as the slope (n) and y-intercept was the value of ln (k). The experiment procedure is as follow:

Experimental Procedure:
(1)To get the motion and velocity, we chose to drop coffee filters from the balcony of the Design and Technology building 13 and captured videos of its fall.

(2)We dropped one coffee filter at first, and kept stacking it until it was a total of six coffee filters.

(3)Once we got the videos of the fall of coffee filters, we used Logger Pro to analyze its motion and velocity through the video.

(4)We filled the data into the Microsoft Excel and let the rest calculate it. We created 6 spreadsheets for each trial, increasing mass according to one coffee filter, two, three till six coffee filters.

Equations to put into Excel

mass of 1 coffee filter=0.000872 kg

mass of 2 coffee filters=2*mass of 1 coffee filter=0.001744 kg

mass of 3 coffee filters=3*mass of 1 coffee filter=0.003488 kg

mass of 4 coffee filters=4*mass of 1 coffee filter=0.002616 kg

Mass of 5 coffee filters=5*mass of 1 coffee filter=0.00436 kg

mass of 6 coffee filters=6*mass of 1 coffee filter=0.005232 kg

(5)Using the position and time we got from the video, we graphed the position versus time graph, which we derived as below.

(6)We derived it into natural log formula, so that we could graph it as y=mx+b, in which we knew the slope of the position versus time graph gives us the velocity.

(7) From our derivation, we knew that n was the slope or terminal velocity and y-intercept was ln(k).

Position Vs. Time graph of 1 coffee filter
velocity(slope m)=n=-1.283m/s

Position Vs. Time graph of 2 coffee filters
velocity(slope m)=n=-1.683 m/s

Position Vs. Time graph of 3 coffee filters
velocity(slope m)=n=2.109 m/s

Position Vs. Time graph of 4 coffee filters
velocity(slope m)=n=2.409 m/s

Position Vs. Time graph of 5 coffee filters
velocity(slope m)=n=2.695 m/s

Position vs. Time Graph of 6 coffee filters
velocity(slope m)=n=2.785 m/s

(7) Our model predicted the terminal velocities of various coffee filters as below:

(8) The velocities resulted from our model and the position versus time graphs from the video analysis came out to be pretty close.

Comparison of Velocity from Graph Vs. Velocity from Model

(9) Next was to plot the air resistance force(mg) vs. velocity(v) graph using power fit of Logger Pro. The graph would come out to be a concave up graph, which I forgot to screenshot. When you got the graph and power fit, there would be values of A(the value of k) and B(the value of n). Our recorded value of k was 0.00487 and the value of n was 2.245.

The graph of F resistance vs. Velocity should look something like this.

(10)So, I used Excel to come up with similar graph that represents the air resistance force(mg) vs. velocity graph. In this case, it is ln (mg) vs. ln(v). From our derived equation of ln(mg)=n*ln(v)+ln(k).

Ln(mg) Vs. ln(v) Graph

Conclusion: Comparing the velocities of coffee filters from our model and graphs, it came out to be pretty close to each other. The air resistance force from our graph came out to be F_resistance=0.00487v^2.245.