Modeling the fall of an object falling with air resistance

Objective:  

       The purpose of the lab is to determine the relationship between air resistance force and speed as well as to model the fall of an object including air resistance.

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

The set-up:

       Before modeling the fall of an object including air resistance, we needed to find an equation that express the relationship between air resistance force and speed. Air resistance force on an object should depend on velocity upward for falling object, thus we predicted a equation relating F_{resistance  and velocity.} 

Figure 2: Running the first trial to know how to capture a video. 

F_{resistance =kv}ⁿ

         ^{As we had this equation, we started to do experiment to check whether our prediction was correct or not. To do this experiment, we needed a 2m meter stick, brown coffee filters, a computer with Logger Pro, and some tape to help set up equipment easier. We did this experiment in building 13 (Figure 1), but before that we ran a first trial in class to know how to capture a video and obtained data (Figure 2)} 

       

Figure 1: Here is where we did this experiment

       Since we figured out how to capture the video and obtained data, we moved to building 13 to conduct the experiment. We would drop the coffee filter at the balcony of building 13, then captured the video, and analyzed the data. The coffee filters were added up to five in total. In other words, we did five trials in total with an increasing amount of coffee filter each time. Below is how we did the experiment. (Figure 3)

Figure 3: We made 2 meter stick be static at the balance, then dropped the coffee filter. Each time we increased one more until we got total five coffee filters.

        After conducting the experiment, we analyzed the data we got. Our purpose is to get terminal velocities for each one by fitting linear portion of the position vs. time graph for each. In other words, the slope of a linear fit position vs. time graph will give us the value of terminal velocities. We found the velocity at a fixed height per time. (Figure 4)

Figure 4: Finding the velocity at a fixed height per time.

         We obtained the terminal velocity for each trial by fitting the linear portion of position vs. time graph. Below is an example how we got the terminal velocity. (Figure 5)

Figure 5: We found the slope of linear portion position vs. time graph for the first coffee filter. As a result, the slope m=0.8155m/s gave us the value of terminal velocity.

        We continued doing the same process for four more trials until we reached five coffee filters in total. Consequently, we would get four more values of terminal velocity corresponding to each time we increase the number of coffee filter. After finding the value of terminal velocity, we needed to figure out the value of force resistance. Because the force of air resistance was related to the mass of coffee filters, we measured the mass of 50 coffee filters in order to minimize the uncertainty in measurement. This is how we calculated the force of air resistance. (Figure 6)

Figure 6: Calculating the force of air resistance corresponding to one coffee filter.

       We had the value of force resistance and terminal velocity, we then entered them into Logger Pro and graphed them to check the pattern of the graph. The pattern of the graph would tell us whether our prediction equation was correct or not.  F_{resistance =kv}ⁿ 

Figure 7: The graph of force vs. velocity for one coffee filter.

       Here is an equation we got to show the relationship between force resistance and velocity for one coffee filter. 

Figure 8: The result of part 1 showed that force resistance did have a relationship with the terminal velocity as we predicted F_{resistance =kv}ⁿ

 ^{Part 2: Modeling the fall of an object including air resistance.}

       ^{The objective of this part is to apply our mathematical model we developed in part 1 to predict the terminal velocity of our various coffee filters.}

  ^{Data set-up:} 

           ^{In order to obtain the terminal velocity, we created six columns in order: time, change in velocity, instantaneous velocity, acceleration, change in position, instantaneous position. Below is how the set-up looked like in Excel.}

Figure 9: Setting up these columns in Excel.

         The logic behind why we set up those column above the way they are is shown here (Figure 10).

Figure 10: The reason why we set up those columns above the way they are in Excel

       

         As we learned from non-constant acceleration lab, when we set  ∆t  smaller, we could get a more accurate result. Therefore, we set up our  ∆t = 0.001s, which doesn't change the result so much. We also noticed that we would obtain the value of terminal velocity when the acceleration was zero. Therefore, after entering the data and performing calculations, we would fill our results down until we reach the  region where the acceleration was zero. The value of velocity correspond to that acceleration would be the terminal velocity which we wanted. Below is an example of terminal velocity for the first coffee filter.

Figure 11: The terminal velocity equal 0.8107m/s as the acceleration reaches zero (One coffee filter).

            We did the same process for 2, 3, 4, and 5 coffee filters and acquired four more values of terminal velocities. Then we calculated the percent error based on the results we got in part 1 and part 2. We used this equation for calculation 

Percent error = [(experimental value - true value)] / true value *100%

The result table is shown below.

Figure 12: Calculating the percent error for each trial.

           Our results were "quite good" as the percent error was within 10%. This means that our results were not too far off from the "true" value. The error could be explained by the uncertainty in measurement, the human error, or the wind in building 13 which may effect our data. 

Conclusion

        In this lab, we learned how to model the fall of an object falling with air resistance. We first determined the relationship between air resistance force and speed and modeled it using Excel. As we predicted the force of air resistance have a relationship with the velocity of the object through an equation F_resistance = kvⁿ , our results proved that our prediction was correct. For example, in part 1, when we calculated force and velocity of one coffee filter, we obtained this equation F_resistance = 0.01348*v^1.886, . For part 2, we acquired the terminal velocity for each trial using Excel, then we calculated the percent error. The average percent error was +0.2921% which was pretty good.