free_fall

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 — free_fall [2019/08/15 14:12] (current)river created 2019/08/15 14:12 river created 2019/08/15 14:12 river created Line 1: Line 1: + ====== 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?
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