11-March-2015: Modeling the fall of an object falling with air resistance

Purpose - We will be able to obtain a relationship between air resistance force and speed. Once a relationship is found we will be using excel to predict the terminal velocity and compare the values.

Part 1 
In this part we will be taking coffee filters and dropping them so that we may develop our terminal velocity by allowing the computers camera to record the time and distance the filter falls. We are going to repeat this process five times each time adding another coffee filter. Once all five videos are recorded we will use logger pro to measure our position and time by placing dots in each video as the filter falls.

Image on right: This is our experimental ground. You see we have to identify a marker so that we can give logger pro a set of data in which to measure the distance in which the coffee filters will fall. Once the marker was established we proceeded to drop the coffee filters as logger pros video function captured each fall.

We repeated dropping the coffee filter five times. Each time adding a coffee filter stacked on one another.

Image below: To obtain position vs. time graph we plotted dots on each time frame in the videos we captured. We followed the coffee filters drop adding each dot to each frame while logger pro calculating the data we needed. Once we dotted each frame and had enough data to work with. We highlighted a good range on the graph from which we will achieve a value for terminal velocity (which is the slope) with respect to the amount of coffee filters we had stacked.

Note: Pay close attention to the slope given. It is negative for all five we fitted but we will end up taking the absolute value of each slope.



We have an expectation that air resistance force on a particular object depends on the object's speed, shape, and the material it is moving through this equation.

To find our value of k and n. We are going to have to calculate a couple things and plot it on to logger pro. Logger pro is great at making graphs and giving out information in which we will translate to finding our k and n.




Image Below: We have a free body diagram of how our coffee filter is while falling. Lets say our downward motion is positive and knowing that our speed is constant. Net F=ma in the y-direction. We have F=mg. We know mass from weighing them and gravity we use 9.8 m/s/s. We calculate our force.

Image below:  Obtaining our terminal velocity and force. We record the data into logger pro which displays a nice graph in which to find k and n. In this case A=k and B=n.

Part 2
We will be developing a mathematical model on excel. The goal is that we will be able to predict the terminal velocity of 1 or more coffee filters.

Image below: For excel we need to formulate an equation for acceleration involving our air resistance force. Notice in the boxed area we find how we will be plugging in an equation for our acceleration into excel.

In excel we will set in our first 5 rows in column A the values our obtain from our calculations. Mass and our time interval will vary. Mass varies by deciding whether it is one filter or five filters we can sum the mass and plot into excel. Time interval varies by how we can view our value shorter in range.

Note: I kept highlighted the box we input our equation of acceleration into.

Once we plotted the necessary data onto excel. We dragged down the columns and allowed excel to calculate the prediction for terminal velocity. 

We know terminal velocity occurs when acceleration reaches zero. Image Above: In purple is acceleration reaching zero and in red is velocity. If you remember from the first graph we obtain from logger pro the numbers are very close to one another.


Image Below: This is a velocity vs. time graph. Notice as time passes our object begins to level off and our velocity is becoming constant. Which give us a max value.

Conclusion

We ultimately learned two ways in which we are able to calculate terminal velocity. First method was through logger pro and its nice video function. Logger pro gave us slopes that meant terminal velocity. We derived a function to use into our acceleration column in excel. Plotted data and knowing that when acceleration reaches zero we have terminal velocity.

With logger pro we had some obstacles that may have infringed on certain values. For example, while plotting dots in the video frames. Visually finding where our coffee filter was a challenge at times. Which may have lead to some error in how logger pro calculates. 

The model in excel gave us a more precise value in terms of our calculation we entered. Any error from excel would have been from the measuring devices for mass of our coffee filter (which effect logger pro as well). Generally most the values we obtained were from logger pro. If any error may have been presence it all began from logger pro.

In the end comparing our values for terminal velocity in logger pro and excel were close enough to validate that both warrant sufficient enough values.