13-Mar-2017: Modeling the fall of an object falling with air resistance

Lab 4: Modeling the fall of an object falling with air resistance

Name: Kevin Nguyen

Lab Partners: Jose Rodriguez, Kevin Tran

Date of lab performed: 13 March 2017

Statement/ purpose: The purpose of this lab was to determine the relationship between air resistance force and speed and to create a model on excel that will be able to determine the terminal velocity of the falling objects that matches our experimental data. 

Theory/ Introduction: In the first part of the lab, we assumed that the air resistance force on an object can be modeled by the equation below.

Air resistance force = kv^n

We used this equation because this equation takes into account the physical characteristics of the falling object when air resistance force is acting on the object. In this equation, "k" is the term that takes into account the shape, material, and area of the object. "n" is the slope of the function and "v" is the velocity of the object. In this case, since we don't know "k" and "n" and "v" is the independent variable, we measured the unknown variables by first video capturing the objects falling inside the design building. In this experiment, we captured 5 instances of the object falling to the ground. 

For each instance, the amount of coffee filters dropped increased by one. So, for example, on drop 1, only 1 coffee filter dropped. On drop 2, 2 coffee filters stacked together dropped. We recorded these videos in order to find "k" and "n". We did this by recording the terminal velocities for each drop. To find the terminal velocity, we set a position vs time graph (where position is in the y-direction and time is in the x-direction) and found the slope of the line fit of the last few points. After finding the terminal velocities, we plotted these points on logger pro and made a power line fit, where terminal velocity is plotted in the x-direction and weight of the coffee filter (m*g) is plotted in the y-direction. The values of "k" and "n" is given from the equation of the power line fit. 

In the second part of the lab, we used the mathematical model of air resistance force used in part 1     (Air resistance force = kv^n) in order to predict the terminal velocity of the falling coffee filters. In order to do so, we need several values. These values are ∆t, the change in time, "m", the mass of the coffee filter(s) dropped, "g", gravity, "k" and "n", the variables defined in part 1, ∆v, the change in velocity, v, the velocity, "a", the acceleration of the falling coffee filters, ∆x, the change in position, and x, the position of the object from the release point. These values are used to make a model on excel in order to predict the terminal velocity of the falling filters, which is determined by the variable "v" and "a", since once acceleration is 0 m/s^2, the velocity will be constant. 

Summary: For part 1 of the lab, we went to the design building in order to record the coffee filters falling from their release point. 

After video capturing the drops, we found the terminal velocity by plotting the position of the falling coffee filter each 3rd of a frame (not pictured). After plotting the points, we made a line model of the last portion of the plot and used its slope to find the terminal velocity. The reason why we made the line model at the end of the graph was because we needed the maximum velocity of the object (maximum velocity = no more acceleration). 

Drop 1

Drop 2

Drop 3

Drop 4

Drop 5

After finding the terminal velocities of each drop, we made another plot on logger pro. We plugged in the values for terminal velocity (before anything, we used the absolute value of these values since the power model can not use negative values) and weight of the coffee filters and got this.

Power model

For part two of the lab, we set up our excel spread sheet similar to this model.

We plugged in our values in the relevant boxes.

After plugging in the necessary values, we set up the variables (t, ∆v, v, etc...) in the spreadsheet so they have the following equations. 

After putting in the following equations into their respective boxes, we "filled" the information down in order to get our answer for terminal velocity.

The highlighted portion represents the terminal velocity of 1 falling coffee filter at ∆t = 0.01 seconds.

We also found the terminal velocities of the 2nd, 3rd, 4th, and 5th drops by changing the mass so it matched their respective drops.

Terminal velocity for the second drop

Terminal velocity for the third drop

Terminal Velocity for the 4th drop

Terminal Velocity for the 5th drop

Table of Recorded Data:

Data collected for the First part

Data needed for the Second Part

Calculated Results:

Calculated "k" and "n" for part 1

Shown below is the calculated terminal velocities (highlighted yellow) for drops 1, 2, 3, 4, and 5 for part 2 of the lab.

Predicted terminal velocity for Drop 1 is 1.50 m/s

Predicted terminal velocity for Drop 2 is 1.97 m/s

Predicted terminal velocity for Drop 3 is 2.31 m/s

Predicted terminal velocity for Drop 4 is 2.58 m/s

Predicted terminal velocity for Drop 5 is 2.82 m/s

Explanation of graphs:

This is explained in Theory/ introduction section.

Conclusion:

The velocities that we recorded from the video capture were slightly different from the values that we predicted using the spread sheet. For drop 1, the velocity prediction had a 1.3% error from the terminal velocity recorded. For drop 2, the velocity prediction had a 2% error from the terminal velocity recorded. For drop 3, the velocity prediction had a 2.5% error from the terminal velocity recorded. For drop 4, the velocity prediction had a 3.7% error from the terminal velocity recorded. For drop 5, the velocity prediction had a 2.7% error from the terminal velocity recorded.

The reason why these predicted values may have had a percent error ranging from 1.3% to 3.7% was because during the video capture, we set the frames per dot to be 3 frames. If it was a smaller value, it may have lowered the percent error of the predicted values since the recorded values may be more accurate.