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+ | ====== Free Fall ====== | ||

+ | ===== Purpose ===== | ||

+ | |||

+ | We have all learned that gravity pulls on things at the same rate. | ||

+ | Therefore, a bowling ball and a feather will experience the same | ||

+ | gravitational acceleration. Seemingly in contradiction, you | ||

+ | instinctively know that if you dropped a feather and bowling ball at the | ||

+ | same time, the bowling ball would land first. Despite being often | ||

+ | ignored in lecture, air resistance is an aspect that can't be ignored in | ||

+ | practice. In fact, it is an important consideration when exploring how | ||

+ | objects move in our world, something no skydiver would contradict. | ||

+ | |||

+ | In this lab, your group is tasked with observing how objects fall and | ||

+ | the ways air resistance affects them. By investigating the concept of | ||

+ | terminal velocity, you will model how an object's maximum speed is | ||

+ | related to its mass. Along the way, you should become more familiar with | ||

+ | the equipment and data analysis techniques you will be using throughout | ||

+ | the semester as well as developing productive skills to work more | ||

+ | effectively in groups. | ||

+ | |||

+ | ===== Theory ===== | ||

+ | |||

+ | In order to investigate the effects of air resistance on an object's | ||

+ | trajectory, it is important to review some important principles. We know | ||

+ | that the force acting on an object can be rewritten as a sum of all | ||

+ | other forces on it. This is an experimental fact, something we observe | ||

+ | time and again in many different experiments. That is, | ||

+ | |||

+ | $${\overrightarrow{F}}_{\text{Net}} = \Sigma{\overrightarrow{F}}_{i} = {\overrightarrow{F}}_{1} + {\overrightarrow{F}}_{2} + \ldots$$ | ||

+ | |||

+ | where${\overrightarrow{\ F}}_{\text{Net}}$ is the total force on an | ||

+ | object and ${\overrightarrow{F}}_{i}$is the individual contribution of | ||

+ | each force. It is important to remember that these forces are //vectors//, | ||

+ | and therefore the direction of each force matters. | ||

+ | |||

+ | From Newton's second law, we know that the acceleration of an object | ||

+ | (//a//) is relative to the mass of that object (//m//) and force acting on | ||

+ | it (//F//). Again, this result comes from many experimental observations | ||

+ | of objects experiences forces. More commonly, we see this written as | ||

+ | |||

+ | $$F = \text{ma}$$ | ||

+ | |||

+ | When considering freely-falling objects, the acceleration that they | ||

+ | experience is //g//. | ||

+ | |||

+ | Air resistance, another force acting on a falling object, can be | ||

+ | considered as | ||

+ | |||

+ | $$F_{D} = \frac{1}{2}\rho v^{2}C_{D}A$$ | ||

+ | |||

+ | where | ||

+ | * $F_D$ is the drag force | ||

+ | * $\rho$ is the mass density of the fluid | ||

+ | * $v^2$ is the velocity of the object | ||

+ | * $C_D$ is the drag coefficient | ||

+ | * $A$ is the area. | ||

+ | |||

+ | |||

+ | By combining these equations, we can determine the acceleration each | ||

+ | object feels as well as the terminal velocity of an object, dependent on | ||

+ | its mass. Take note that the gravitational force and the drag force act | ||

+ | in diametrically opposed directions for objects falling in a straight | ||

+ | line. | ||

+ | |||

+ | ===== Research Concepts ===== | ||

+ | |||

+ | In this lab, like many others this semester, you'll likely benefit from | ||

+ | video tracking and obtaining your data from the videos. As such, prior | ||

+ | to class it's useful to understand: | ||

+ | |||

+ | * What terminal velocity means and what parameters on which it depends | ||

+ | * What a vector means and how they can be combined | ||

+ | * How the above equations can be combined to determine the relationship between mass and terminal velocity | ||

+ | * How you can determine the speed of an object from a displacement vs time and velocity vs time graph. | ||

+ | |||

+ | Additionally, you will be using video tracking software in many labs | ||

+ | this semester, including this one. Therefore, it would be useful to: | ||

+ | |||

+ | * Download video tracking software from [[http://physlets.org/tracker/]] (the computers in the lab have this as well, but it may be useful on your own devices, too) | ||

+ | * Understand how to use the software, especially regarding how to track specific objects and how to analyze data ([[http://physlets.org/tracker/help/frameset.html]]) | ||

+ | * Look up the frame rate of the camera in your phone, as well as what slow-motion options it has (and the frame rate for any slow motion functions on your phone). | ||

+ | |||

+ | ===== Tracker Tips ===== | ||

+ | |||

+ | Throughout the semester, you will be expected to make decisions with | ||

+ | your data and apparatus when conducting experiments. However, because | ||

+ | this is the first time you will be using the video tracking software, we | ||

+ | wanted to share some tips to help expedite your data acquisition and | ||

+ | analysis. This list is not exhaustive, and complications in an | ||

+ | experiment can arise unexpectedly. However, these common issues can be | ||

+ | avoided through thoughtful experimental design: | ||

+ | * Pay attention to your surroundings, ensuring that there is enough contrast between the falling object and background, especially if the background is in focus. | ||

+ | * Many videos will look the same, so finding a way to designate between them will expedite analysis. | ||

+ | * Consider a way to calibrate parameters on the video, especially distance. | ||

+ | * Ensure your camera is being held still | ||

+ | * Try taking and analyzing a test video before taking all of your data. You may determine some issues with your setup that you can fix before it's too late. | ||

+ | |||

+ | ===== Free Fall Experiment ===== | ||

+ | |||

+ | **Part 1 -- Determining "g" from a Free-Falling Object** | ||

+ | |||

+ | You all know that letting go of a carried object will cause it to fall | ||

+ | due to gravity. However, using video-tracking software, we can obtain a | ||

+ | value for the acceleration of the fall, or "g." With your group, choose | ||

+ | an object to drop, recording the fall with a camera (i.e., your phone). | ||

+ | |||

+ | //You are responsible for your equipment, so make sure the object you | ||

+ | choose will not break.// | ||

+ | |||

+ | Obtaining valuable data will require participation from the entire | ||

+ | group. There are many aspects to consider while conducting this | ||

+ | experiment, so determine with your group who will be responsible for | ||

+ | each aspect in order to conduct your experiment efficiently. When | ||

+ | recording this free-fall, consider: | ||

+ | * The equation you are using to model the object's motion | ||

+ | * What parameters you will need to know or measure (i.e., distance, time, mass, etc.) and how you will be obtaining them from the video or the data? | ||

+ | * What sources of uncertainty you are considering and the relative effect of these sources? | ||

+ | |||

+ | From your video data, determine the acceleration of the object. | ||

+ | * How does it relate to the "known" value of g, 9.81 m/s^2^? | ||

+ | * Can you account for any differences between your value and the "known" value? | ||

+ | |||

+ | **Part 2 -- Observing Drag ** | ||

+ | |||

+ | You just observed what happens when dropping a bulky object, but as you | ||

+ | know intuitively, a bowling ball and a feather don't fall at the same | ||

+ | rate. Therefore, an object's properties must be a factor determining how | ||

+ | fast it falls. We can observe this by tracking an object we know will | ||

+ | fall differently, like a coffee filter. | ||

+ | |||

+ | Drop a coffee filter from an appreciable height and watch how it falls. | ||

+ | When making observations of the falling filter, consider: | ||

+ | * How does the filter fall? Why is this so different from the object dropped in **Part 1**? | ||

+ | * Does the way the filter fall depend on how it is dropped? Consider dropping the filter with different orientations to draw conclusions. | ||

+ | * Are there ways you can design your experiment to maintain consistent orientation during the fall? | ||

+ | * Is there a minimum height you can drop the filter from to make sure it reaches terminal velocity? | ||

+ | |||

+ | **Part 3a -- Determining Terminal Velocity ** | ||

+ | |||

+ | When you are ready to take quantitative data, record the motion of the | ||

+ | filter as it falls, using the video tracking software to help analyze | ||

+ | your data. How you determine the terminal velocity from your video will | ||

+ | be up to you and your group, but keep in mind your variables and the | ||

+ | benefits of the tracking software, such as the graph and data tables. | ||

+ | (Keeping these in mind will help with the rest of the experiment.) While | ||

+ | analyzing your data, it would be useful to consider: | ||

+ | * How are you determining and measuring the terminal velocity? | ||

+ | * How confident are you that the filter has reached terminal velocity? | ||

+ | * How can you use your data to help increase confidence in the value reported as well as decrease the uncertainty? | ||

+ | * What might happen to the terminal velocity if you stack multiple coffee filters? | ||

+ | |||

+ | **Part 3b -- Determining the Relationship Between Mass and Terminal | ||

+ | Velocity** | ||

+ | |||

+ | By stacking filters, you can change the mass of the object without | ||

+ | adjusting the shape (i.e., your drag coefficient and area remain | ||

+ | constant). That way, you can investigate how the terminal velocity is | ||

+ | related to the mass of the object without changing any of the other | ||

+ | variables in your equations. | ||

+ | * While adding coffee filters, is there a point at which terminal velocity is no longer observable? | ||

+ | * If so, can you adjust your experiment in order to still measure this? Think of all the variables in the equation and in your experiment (i.e., those not necessarily in the equations). | ||

+ | * If you can no longer determine terminal velocity, why not? | ||

+ | * How many different masses are you able to test before you can no longer determine terminal velocity? | ||

+ | |||

+ | **Part 4 -- Synthesizing Your Data** | ||

+ | |||

+ | You can determine the terminal velocity of each individual video using | ||

+ | the tracking software. In order to relate each trial, you will have to | ||

+ | use Excel (or similar software). Transfer your data into Excel and | ||

+ | determine how terminal velocity depends on mass. When modeling data, it | ||

+ | is often helpful to represent the data graphically. When creating your | ||

+ | graph, consider: | ||

+ | * Under what parameters does your plot become linear? | ||

+ | * How does this relate to the theoretical equations given? Does your data support theoretical models? Why or why not? | ||

+ | * If so, can you determine any quantitative information from your plot? (When modeling, the slope and intercept are often useful values.) | ||

+ | * If not, why not? What factors make the relationship difficult to determine? | ||

+ | * Are you able to conclusively determine anything from your data? If not, what would you need to be able to draw conclusions? | ||

+ | |||

+ | ===== Questions to Think About ===== | ||

+ | |||

+ | As you conduct your experiment, it may be helpful to consider: | ||

+ | * How are you assigning your uncertainty? | ||

+ | * Are there ways to design your experiment so that you minimize your uncertainty? | ||

+ | * What is your goal for each part? Have you considered how you will analyze your data, ensuring your design will be appropriate? | ||

+ | * How are you going to determine when the filter moves at terminal velocity? |