In the introductory lab, we made our own measurement device and used it to estimate the length dimensions of a card. When making these measurements, we had to estimate uncertainties on each measurement, and most students determined those uncertainties based on the limited resolution of their ruler. This week, we want to explore a completely different type of measurement – one where the uncertainty is better estimated through *repeated measurements* and *statistics*.

In the following project, you will perform some simple experiments – dropping slips of paper from fixed heights – but we will use this data to develop ideas about experiment design, repeatability and systematic effects, statistical distributions, and model-testing. At the end of the experiment, you will use the physics you've learned so far this quarter – kinematics, forces, terminal velocity, and drag – to try to interpret your results.

In this experiment, you will investigate the lateral displacement of a piece of paper as it falls through the air under the force of gravity. See, Fig. 1 for an illustration.

Why would we want to study this?

When dropped, many objects such as balls, coins, or laptops simply fall *straight down* until they hit a point directly below the drop point; this is the type of motion that many of your homework problems about acceleration due to gravity are based on. However, if a piece of paper – something light and with a large surface area – is dropped, it is not likely to drop straight down. Instead, it will flutter about randomly as it falls and ultimately land some distance away from the initial drop point.

Calculating how far from the drop point the paper will land is a complex physics problem… and one well beyond the scope of this course. But, if we look at enough paper drops, we may be able to discern patterns which can give us some insight into the overall behavior. Experimental physics often takes this form: *trying to design a simple measurement of a physically complex phenomenon.*

We have chosen this particular experiment for our first lab project for a number of reasons.

- It is easy to perform at home using commonly available items – namely paper, scissors and a ruler or tape measure.
- It is a phenomenon for which there is no known “right” answer which you can look up to see if you did it correctly. (Physics experiments are rarely performed for the purpose of confirming something you already know.)
- The data are well suited for statistical analysis and estimation of uncertainties, and as such, this experiment provides excellent continued practice with the techniques we will use all year for determining how well you know something you have measured.
- The variety of choices that you will make as you design your experiment – and the importance of being able to describe and defend those choices later to the group – illustrates the importance of maintaining a good scientific notebook.

The experiment will be performed in two phases.

- First, each member of the group will conduct an
**individual experiment:**you will create paper squares, drop them from a fixed height, and measure the displacement. Afterward, you will meet with your group members and your TA via Zoom to compare results and to design an experimental procedure which will be used to repeat the measurement for different heights. - Second, each member of the group will perform part of the
**group experiment:**the group will investigate lateral displacement as a function of drop height. During this part, it is important that you complete all your tasks on time and according to the method the group has agreed upon because your data will form part of a larger set. You will again meet with your group members and the TA to discuss the final results.

You will record all the work you do in your electronic lab notebook using the provided Google Docs template. It will be important to record notes on your experimental setup, the data taken, any calculations and analysis performed, and discussion notes and conclusions from group meetings.

The goals for this first experiment are the following:

- to introduce students to…
- …designing an experiment and being able to explain their procedure to others;
- …using statistical tools such as the mean, standard deviation, and standard deviation of the mean (or standard error) to analyze data and facilitate the comparison of results; and
- …collaborating as a group towards a common goal;

- to give students an experimental question for which there is no known right answer; and
- to use group discussions with TAs to…
- …illustrate how uncertainties are used to determine if results are in agreement; and
- …highlight the importance of lab notebooks as a record of experimental procedures and results.

*This section is to be completed individually BEFORE your first meeting with the TA and your lab group.*

As with the previous lab, we provide a lab template for you to use as your lab notebook. This time, you will notice that the prompts are more open-ended. You will need to decide what information to record and present!

Remember to fill this out **as you go along**. Do not wait until you have completed the experiment. The notebook is meant to be a record of everything you've done in lab – good and bad – and is only useful if it is complete and honest.

Your individual task is to measure the lateral displacement of pieces of paper dropped for a fixed height.

- From a piece of standard-weight printer or notebook paper, cut out a number of square pieces,
**1 cm on each side**. - Drop each piece of paper from a
**height of 60 cm**. **Measure the distance**from the spot directly underneath the drop point to each piece of paper. Do this for at least 20 drops. (You may do more drops if you wish.)- Record all of the
**important details**of how you set up and performed the experiment, including photos and/or diagrams.

Keep in mind that at your first group meeting, everyone will share details of how they performed their measurements. During this meeting, the group will discuss how to determine if everyone is in agreement and will collaborate on designing a follow-up experiment. Be prepared to discuss your results and your methods!

As you perform your experiment, try to record everything that might impact your measurements. For example, consider the following questions:

- How did you release the pieces of paper to ensure that they initially had no lateral velocity?
- How did you determine the location of the point on the floor directly below the release point?
- How did you measure the height of the release point?
- How did you measure the lateral displacement?

The above is not an exhaustive list. Think of other questions, and record those details in your notebook as well.

NOTEBOOK: Summarize the observations you make as well as the procedure you chose. Include photos, drawings, and/or sentences to describe your setup and what you see.

NOTEBOOK: Record your data. What format should it take?

Did you find that every single paper drop landed the same distance from the center? (*Probably not.*) Does that mean that you did the experiment incorrectly? (*Probably not.*)

Many things we measure in physics depend on small random fluctuations. In this case, the exact path of the paper as it falls down depends on every interaction it has with the air molecules… or with the influence of tiny air currents… or with the exact angle of release… or whether the paper stuck to your thumb for 0.2 ms or fell off immediately… *and so on*. These many small effects mean that it is impossible to predict the path of a single dropped piece of paper, but we may still be able to say something about the behavior of *many* dropped slips.

If you drop a single slip of paper, it is difficult to tell if the distance it lands away from the center is typical or not. But as you collect more and more drops, you can begin to see clustering of values. You may find, for example, that a displacement of 18 cm might seem *typical*, whereas a displacement of 75 cm or 1 cm seem *atypical*.

How do we quantify this?

Your TA will talk about ** distributions ** of values during your group meeting and will help to explain the following terms more carefully, but for now, let's define some useful statistical measures. For the following, we will assume that you dropped your paper $N$ times and measured a list of displacements $x = [x_1, x_2,..., x_N]$.

- The
**mean**(or**average**) of a list of numbers is given by $\mu = x_{\textrm{avg}} = \frac{1}{N}\sum_i^N x_i$. - The
**standard deviation**of a list of number is given by $\sigma = \sqrt{\frac{\sum_i^N (x_i - \mu)^2}{N-1}}$. - The
**standard deviation of the mean**(or**standard error**) is given by $\sigma_{\mu} = \sigma/\sqrt{N}$

We'd like you calculate these three values for your list of displacements before coming to the meeting. (Don't worry too much about *why*… your TA will explain more.)

To help you with these calculations, we provide an online program here. This is a Google Collaboration notebook running Python code. **You do not need to know how to code to use the notebook… it is just a tool!**

If you are unable to get the notebook working, ask you TA for help in the meeting. In the meantime, you can use this site instead.

NOTEBOOK: What was the mean displacement of your slips of paper? (The standard deviation? The standard deviation of the mean?) Are there any outliers that might have affected your results?

Remember to submit your lab notebook **before** your group meeting!

At this point you have performed an experiment which produced a measured value and an estimate of how well you know that value.

In the second part of this project, your group will investigate the relationship between the lateral displacement and the drop height. Each person in the group will measure the displacement for a different height, and these measurements will be combined into one data set for everyone in the group to use. In order for these measurements to be useful, you need to be certain that everyone is using the same procedure so that any one person's result could be reproduced by another member of the group.

Your TA will lead a group discussion which should roughly cover the following points:

*Are your individual measurements of lateral displacement for 60 cm in agreement?* This is where your error analysis and uncertainty estimate comes into play as it provides an objective standard for determining how close two numbers have to be to be considered in agreement with one another.

*If you are not all in agreement, what differences in methodology are there which might be causing your results to be different?* This is where your notes on how you set up and performed your individual experiment come into play. You will have to compare details of how the measurements were made. Did some people hold the pieces of paper vertically before dropping while others held them horizontally? Did some in the group measure the displacement from the drop point to the nearest edge of the piece of paper while others were measuring to the far edge? Was someone's cat batting at the pieces of paper as they were falling? There are many small details which can cause two people's measurements to disagree. The more detailed and complete your notes on your experiment, the easier it will be to identify potential sources of discrepancy in the data.

*What procedure will be used for the next set of measurements to ensure that all the data are consistent?* Based on what you learn from the previous discussion, develop a detailed procedure that everyone will follow in the next part of the experiment. These details need to be recorded in your lab notebook so that when you do take the data you can be sure you followed the proper procedure. Think about how you might confirm whether everyone's results are consistent.

This exercise is designed to illustrate how scientists work together to make independent measurements of the same phenomena with confidence that each team member is actually measuring the same thing.

NOTEBOOK: Take notes during your meeting. What did your group talk about? What results (from you or your groupmates) are important to keep in mind? What information did the TA provide to guide you? What did your group decide to do next in Part 2?

*Do not begin on this part until AFTER your first meeting with your group and TA.*

Continue recording your notes and data in the **same notebook** document as before. (Only add to it. Do not delete or edit anything from Part 1. Do not start a new lab notebook.)

Using your notes from the previous meeting, perform your measurements. Follow the agreed-upon procedure as closely and carefully as you can, recording notes on what you did and how you did it. Compute your mean values, standard deviations, and standard errors. Keep in mind that at your next group meeting your data will be combined with the rest of the group's data to form one data set for analysis.

NOTEBOOK: Remember to record your process and your data in your notebook! Make sure to keep detailed notes since you will need to refer back to them at your meeting and when you write your summary and conclusions.

The reason that your group is now collecting data at different heights is that we want to explore how the lateral displacement changes as a function of height. Let's review the idea of terminal velocity and then look at two possible models more closely.

In your homework problems, we normally assume that a falling object is subject *only* to the downward force due to gravity. For many objects, this is a good enough approximation, but for our slips of paper, the flat shape and light weight mean that we cannot neglect air resistance. In this case, our force diagram has two forces – gravity pointing downward and air resistance pointing upward. The magnitude of the force due to gravity is constant, but the magnitude of the force due to air resistance will increase as the object speeds up.

Initially, the force due to gravity is much larger than the force due to air resistance, and the net force (and therefore acceleration) points downward. But as the paper increases in speed, the force due to air resistance also increases and the the net force downward *decreases*. Eventually, the two forces become equal and the net force (and therefore acceleration) goes to zero. At this point, the paper is falling at a constant velocity which we call **terminal velocity**.

In both of our following models, we will assume that our paper *immediately* comes to terminal velocity and therefore the vertical velocity $v_v$ is constant for the entire fall. Since the velocity is constant, the time $t$ that it takes for the paper to fall is proportional to the height $h$ of the drop: $h = v_v t$ . If you double the height, for example, it will take twice as long to fall down.

Suppose that when you drop the paper, it has some small non-zero initial horizontal velocity, $v_h$. Maybe there is a small random angle on each drop, or maybe the shape or curve of the paper gives it an initial boost in a particular direction when you let go. Either way, this initial velocity will mean the paper continues in the direction of that initial velocity at that constant speed for the entire time of the fall. Since the time for the fall is proportional to the height, and because the horizontal velocity is constant, the lateral displacement of any individual slip of paper $d_i$ is proportional to the height: $d_i = v_h t = v_h (h/v_v)$. The initial direction of the horizontal velocity may change from drop-to-drop and the magnitude of the initial horizontal velocity may fluctuate a bit from drop-to-drop, but averaged over many drops we would expect the magnitude of the average lateral displacement $d_{avg}$ to be proportional to the height.

Model 1: Average lateral displacement is proportional to (i.e. linear in) height: $d_{avg} \propto h$. |

Suppose that when you drop the paper, it doesn't have any initial horizontal velocity, but instead just flutters left and right or forward and back as it falls as it interacts with the molecules of air. If the direction of motion is random – sometimes the air pushes it one way, sometimes another, with no discernible pattern – then this is called a **random walk**. An object undergoing a random walk tends to move away from the place where it started, but it does so *slower* than an object going in a straight line. (Sometimes the object moves *away* and sometimes it moves *back.*)

Random walks are a well-studied phenomenon in physics. We won't derive this formula, but an object undergoing a random walk will have an average displacement that obeys the relation $d_{avg}^2 \propto t$. Again using the fact that fall time is proportional to height, we have now that the magnitude of the average lateral displacement is proportional to the square root of height.

Model 2: Average lateral displacement goes as the square root of height: $d_{avg} \propto \sqrt{h}$. |

Again, submit your *(updated)* notebook **before** the meeting.

Before the meeting, share your results with the rest of the group (by email or Canvas message). If you have time, try to plot the data yourself to see how it looks. (The Google Colab notebook from above includes a place to do a quick plot in Part 2 here.)

At the meeting, the TA will collect your data, plot it, and lead a discussion on the results. Be prepared to discuss your methods and share data, and be ready to participate in a discussion about how your group's data compares to the two potential models.

After your second meeting, you will again need to write up your summary and your conclusions. Include any data tables, plots, etc. from the experiment or discussions as necessary in order to show how your data support your conclusions.

This part doesn't need to be long; one or two pages should be sufficient. What is important, however, is that your writing should be complete and meaningful. Address both the qualitative and quantitative aspects of the experiment, and make sure you cover all the “take-away” topics in enough depth. Don't include throw-away statements like “Looks good” or “Agrees pretty well.” Instead, try to be precise.

Remember… your goal is not to discover some “correct” answer. In fact, approaching any experiment with that mind set is the exact wrong thing to do. You must always strive to reach conclusions which are supported by your data, regardless of what you think the “right” answer should be. Never, under any circumstances should you state a conclusion which is contradicted by the data. Stating that the results of your experiment are inconclusive, or do not agree with theoretical predictions is completely acceptable if that is what your data indicate. Trying to shoehorn your data into agree with some preconceived expectation when you cannot support that claim is fraudulent.

**REMINDER**: Your report is due 48 hours before your next meeting. Submit a single PDF on Canvas.

Finally, lest you think that experimenting with falling pieces of paper is somehow too trivial to be considered **real** physics, check out these articles from research journals on the subject!

Below are some resources for TAs to use. If you're a student… you can look, but you don't need to go through or understand anything here (unless your TA asks you to explore any of these things during your meeting.)

When we make observations of something which varies with random fluctuations, that data typically obeys a **Gaussian** (also called a **Normal**) **distribution**. Such a distribution can be characterized by parameters like the *average,* $\mu$, (or *mean*) and *standard* deviation, $\sigma$. The standard deviation tells us something about the typical “spread” in values. It tells us how wide the distribution is.
Suppose we have a hypothetical experiment where we measure the _time it takes for a drop to fall from a given height. The figures below which show a Gaussian distribution evolving as we add more and more data.

As the number of drops increases, the distribution looks more and more like a smooth Gaussian and we get better estimates for both the average and the standard deviation. But notice that the standard deviation (the width of the distribution) doesn't change much; taking more data doesn't make the distribution more narrow.

For that reason, we need to define a different parameter… one that will scale as the number of data points increases to reflect the fact that our estimate of the uncertainty also improves with the number of counts. This parameter is called the **standard deviation of the mean**, $\sigma_{\mu}$, (or, sometimes called the **standard error**):

$\sigma_{\mu} = \sigma/\sqrt{N}.$ |

If we look now at those data distributions again, we can see that the standard deviation of the mean *does* decrease as we add more data. Hence, we can now more carefully state our average and its uncertainty as $\mu \pm \sigma_{\mu}$.

Using the plot with 500 drops, we can now say that the average fall time is T = 0.997 +/- 0.006 seconds.

When measuring real-world phenomena, most events do not produce exactly the same outcome every time.

For example, consider how long it takes to get to this class. How long does it take? How precise can you be? Unless you can clear the entire path through campus every time you commute, there will probably be a few outside factors (traffic, construction) that will affect your trip. Even taking those into account, your pace will not be identical from day to day; you may stumble on the stairs, or any number of things that aren't controllable can affect the outcome. However, if you timed your commute daily, you would likely find that you could confidently put a range of times on your commute which is representative of typical circumstances and be confident making plans based on that range.

The activity we just did illustrated one way of assigning uncertainties to measurements using statistical fluctuations. In other instances (such as measuring static objects), the uncertainty of a measurement may be determined by the *resolution* of your measuring device. If your stopwatch only measures time with 0.01 s resolution, then you can't expect to make very accurate measurements of shorter lengths of time.

Finally, there are instances where the *relative uncertainty* in a measurement is important. Relative uncertainty is just the absolute uncertainty of a measurement divided by the measurement value. For instance, the relative uncertainty in $A$ is $\dfrac{\delta A}{A}$.

** Propagating uncertainties **

In many instances, the thing we want to measure isn't directly accessible. For example, there's no device that can measure kinetic energy directly. However, since we know that kinetic energy depends on the mass and velocity of an object, we can derive it by measuring those separately. Other times, a direct measurement would just be less practical. For instance, if you wanted to know how far the top of the Sears tower is above sea level $H$, you would want to measure the height of the tower relative to the ground $h_1$, and then add it to the elevation of Chicago above sea level $h_2$. In order to have some expression for how confident we are in our calculated values, we need to be able to combine (propagate) the uncertainties in our measurements somehow. For most situations, the following three rules will be sufficient for this course:

** Numerical constants have no uncertainties **

The radius of a circle $r$ is equal to half its diameter, $r = \frac{1}{2}d$. The uncertainty $\delta r$ is then just $\delta r = \frac{1}{2} \delta d$
** If a quantity is a sum or difference, then its uncertainty is the the square root of the sum of the absolute uncertainties of its components squared **

Suppose we're finding the height of the Sears tower,$H = h_1 + h_2$. The uncertainty in $H$ is expressed as $\delta H = \sqrt{(\delta h_1)^2 + (\delta h_2)^2}$

** If a quantity is a product or quotient, then its relative uncertainty is the square root of the sum of the square of the relative uncertainties of its components squared **

Example: If we're finding the area $A = lw$ from the length $l$ and width $w$ of a rectangle, and the relative uncertainty $\dfrac{\delta A}{A}$ is expressed as $\dfrac{\delta A}{A} = \sqrt{\left(\dfrac{\delta l}{l}\right)^2 + \left(\dfrac{\delta w}{w}\right)^2}$.

**Example:**

In the case of kinetic energy, we have $E_k = \frac{1}{2}mv^2$. This can be broken down into a product of four terms, $E_k = \frac{1}{2} \times m \times v \times v$. The relative uncertainty $\dfrac{\delta E_k}{E_k}$ is expressed as $\dfrac{\delta E_k}{E_k} = \sqrt{ \left(\dfrac{\delta 1/2}{1/2}\right)^2 + \left(\dfrac{\delta m}{m}\right)^2 + \left(\dfrac{\delta v}{v}\right)^2 + \left(\dfrac{\delta v}{v}\right)^2}$ The fraction $\frac{1}{2}$ is exact, so it has no uncertainty: $\delta\frac{1}{2} = 0$ . With this, we can simplify the relative uncertainty as $\dfrac{\delta E_k}{E_k} = \sqrt{ \left(\dfrac{\delta m}{m}\right)^2 + 2\left(\dfrac{\delta v}{v}\right)^2}$.