Mar 11th, 2015: Lab 4; Modeling Fall of an Object with Air Resistance

Purpose: 
We want to measure the terminal velocity of a falling object, and to propose a model to show a relationship between air resistance and weight of an object.

Equation: Fair resistance = kV^n

Experiment:
Part 1
_ The actual experiment took place in Bld. 13. Coffee filters are dropped from the balcony, each time, adding one more coffee filter to the total mass. LoggerPro records the video in 30 frames/sec. We had to manually change the height in the video and pinpoints as dots the position the coffee filter falls into. By doing this, LoggerPro can gather the data and plot them into a position vs. time graph, as shown:

example of x vs. time for 1 coffee filter

_ mass coffee filter= 0.000926 kg
_ The terminal velocity is obtained from the slope (m) from doing linear fit.
_ After 5 terminal velocities are collected, we calculate the Fair resistance. The assumptions here are:

• If coffee filter moves twice as fast, air resistance is twice as much: Fair resistance ~ V
• If the velocity moves as it's multiply by itself and air resistance stays the same: Fair resistance ~ V^2
• If the first 2 are to fail, we must add variable to the equation: Fair resistance = kV^n. As "k" accounts for the sensitive value in surface area and volume of filter when falling.

_ Assuming that at some point during the fall, resistance force= mg, we can label the y-axis below as so. The x-axis is composed of 5 terminal velocities previously collected.

Power law fit used here to find "k" and "n"

Part 2
_ To assure our video capture's accuracy, we also use numerical analysis on Excel.
_ The usual setups are required here as shown below:

Cells setup for 5 coffee filters (only the mass change for each setup); time increment= 1/50

_ From the third assumption (from part 1) and Newton's second law: Fnet: mg-Fair resistance = ma; a=g-[(kV^n)/m].

_ This is why "k" and "n" are needed from part 1. (as shown in the cells setup above)

_ We start with a time increment of 1/30 since that's how much frames the video captures each second. Then we change it to 1/50 just for sake of accuracy.

These two screenshots are from the same trial (mass of 5 filters)

_ Notice how acceleration goes to zero and the velocity stays at 1.88666m/sec. For part 1's value, for the same amount of filters, terminal velocity= 1.8680.  Percentage error ~1%.

Conclusion:

 The above comparison turns out to be the best among the five, and not to mention the propagated uncertainty in mass measurement: dm= +/- 0.1g, error in LoggerPro manual input, and video capture. However, our data gives values of "n" and "k" that are very close to our second assumption. All in all, the quality of lab equipments still prevent us from getting a more suitable data for our air resistance and velocity relationship.