free_fall

Free Fall

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.

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.

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).

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.

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?