13-March-2017: Lab 4: Modeling the fall of an object falling with air resistance

Eric Chong

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

Lab Partners: Lynel Ornedo and Harvey Thai

Part 1: Determining the relationship between air resistance force and speed.

Purpose: The goal of this part of the lab is to capture videos of coffee filters falling from the balcony with air resistance and model a position vs. time graph of the fall.

Introduction/Theory: We modeled the fall of objects without air resistance before, but this time we want to take into account the air resistance that affects the fall. Supposedly, the equation of the force of air resistance changes with respect to the speed of the fall. We can model this in the form of a power law:

Fresistance = kv^n

where F is the force, k is the constant value that takes into account the shape and area of the object and n is the number that determines the shape of the graph. It is suspected that n should be 2, because the gravity force is still accelerating the speed of the coffee filter, until the force of air resistance brings the fall to constant velocity. It is at this point that the force of air resistance is equal to the force of gravity (mg), since the object falling is at equilibrium.

Procedure: First we went to building 13 and came across a balcony that is high enough to model the fall. Here is a picture of a picture of the building and balcony:

We have our laptops at hand and then the professor went up to the balcony and set up the coffee filters. Here is a picture of one of those coffee filters:

We used LoggerPro in our laptops and captured a video of the professor dropping a coffee filter from the balcony. Each time, professor would increase the number of coffee filters stacked on top of each other in order to increase its mass, and we would notice that the fall would start off faster each time. We recorded videos for 1, 2, 3, 4, and 5 coffee filters falling from the balcony. We also recorded one with 6 coffee filters just for good measure.

We then used LoggerPro and tried to plot the points on the video at which the filters' positions are each frame. By doing so, we are also graphing the position of the coffee filters over time. We did this for all of the videos and found the terminal velocity by taking some of the last few points where they are the straightest and used a linear fit and found the slope. Here are the results:

We then took the slopes of all of these graphs and used them in another graph. We made the y-axis of this new graph the force of air resistance (N), and the x-axis the terminal velocity (m/s). We know the mass of the coffee filters by measuring its mass and gravity is given, so we multiplied them by each other in order to get the force of gravity, which is also equal to the force of air resistance at terminal velocity. We graphed the values and did a power fit and here are the results:

On the graph, A is the value of k, x is the velocity, and B is the value of n. In this case, k is 0.006433 +/- 0.0005217, and n is 1.994 +/- 0.08755, really close to 2 as expected.  Now we can move on to part 2 of the lab.

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

Purpose: The goal here is to apply the mathematical model we developed in part 1 to predict the terminal velocity of our various coffee filters.

Procedure: For this part of the lab, we used excel to set up a spreadsheet with this type of layout:

Note that delta x and x are not included in the layout, as the point of the spreadsheet is to find out the values up to acceleration. Like the previous lab, we made t depend on delta t and worked from left to right from there. Here is the general way to set up the cells of each column:

After we set everything up and we inputted the cells with the correct equations, we finally inputted the values and filled down the cells. We then examined the values of v, which is essentially our value of velocity at time t, and we searched for the point at which v is unchanging. This point is our terminal velocity. We did this for all 6 of our graphs from part 1. Here are the results for all 6 of the graphs:

We noticed that the values of v are slightly off from the values we measured by graphing, but they are still relatively close to one another. This concludes part 2 of the lab.

Conclusion: Overall, the lab documented the fall of an object that is subject to air resistance. We modeled the fall of that object and noticed that with an increased speed, there is an increase in air resistance. So, while stacking the coffee filters, we are also increasing the speed of the fall, since the force of gravity increases. This also increases the force of air resistance until it equals out the force of gravity and causes the filters to fall at constant velocity. We then plotted the position vs. time graph and found the terminal velocity. We can also calculate the force of air resistance by calculating the force of gravity. We then graphed the values in a force of air resistance vs. terminal velocity graph and found the values of k and n. This is how we found the equation of the force of air resistance.

One of the uncertainties in our data collection was the video recording. The video recording was poor quality and when we analyzed the position of the coffee filters, sometimes the filters would blur out as its speed increased, causing us to have difficulty pinpointing where its position was. This may have caused our our uncertainty in our points, as some of the points of the position vs. time graph are not linear at terminal velocity. This may have affected our slope, and thus affect the points on the force of air resistance vs. terminal velocity graph.

Overall, the lab teaches the concept of air resistance and how speed affects the magnitude of it. We now know that air resistance can equate to the force of gravity while objects are falling, and we can use this in our future calculations.