This is the make-up lab assignment for students who missed an in-person lab session this quarter. This lab can be done remotely (no in-person activity required) and it is to be completed individually. It should take you approximately 2-3 hours to finish, but it does not need to be completed in one sitting.

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.

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 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 – illustrates the importance of maintaining a good scientific notebook.

The experiment will be performed in two phases.

- First, you will conduct a
**single experiment**: you will create paper squares, drop them from a fixed height, and measure the displacement. You will then compare your results with two “imaginary lab partners”. - Second, you will
**test models**by looking at lateral displacement as a function of drop height. You will extend your measurements to higher or lower heights, and you will compare your data with two possible models which might explain the data.

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 your discussions/conclusions.

The goals for this experiment are the following:

- to introduce students to…
- …designing an experiment and being able to explain their procedure to others; and
- …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

- to give students an experimental question for which there is no known right answer; and
- to illustrate how uncertainties are used to determine if results are in agreement.

Click the link below to create a copy of the template you will use for this first lab. You will be prompted to log into your UChicago account (if you aren't logged in already) and it will ask you if you to create a copy of the template in your personal Google Drive.

We will try to call out places where you need to write in your lab notebook by using the **NOTEBOOK** tag, like in the following:

NOTEBOOK: Fill out the top portion of the first page.

You should write down additional things in your notebook whenever you think it is useful – whether or not we specifically ask for it. These can include notes to yourself (e.g. to record/remember what you've done) or to your reader (e.g. to communicate an outcome or discuss a result). As the course goes on, you will begin to develop a feeling for what needs to go in your notebook and we'll use fewer reminders.

**REMEMBER**: You are going to be graded on the *completeness* of your lab notebook, not how pretty or perfect or correct it is. It should be neat (if the TA can't follow your work, they can't tell if it's complete), but you don't need to labor over it to make it perfect. It is a working document that will help you in meetings and as you write your conclusions for the final report.

Your first 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.

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?

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 $x=[x_1,x_2,\ldots,x_N]$.

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

*For more information on how these values relate to each other (or for a refresher of other uncertainty and statistics definitions), check out the expandable sections below.*

Consider the length measurements you made in today's experiment. When you make such a measurement, it is unlikely that the value you find is exactly on a tick mark of your ruler. And even it looks to your eye like it's on that tick mark, it may not be *exactly* that length. (If you zoomed in with a microscope would it still be perfectly on the line? Is the edge of the card perfectly flat? Is your ruler so perfect that the tick marks are exactly in the right place? The answer to all these questions is “no”.) Therefore, it's not sufficient to simply report a value for a length, you have to also provide your estimate of the *uncertainty* in that measurement.

A **measurement uncertainty** is your best estimate as to how close your “measured” value is to the “true” value. It isn't a random guess or an arbitrary number, but it is judgement you have to make about how much you trust your value. If we say our best estimate for the measured value is $x$ and our estimate for the uncertainty is $\delta x$, then we'd report our final value as $x \pm \delta x$. This does not mean that the true value is *absolutely* within the range from $x - \delta x$ to $x + \delta x$, only that we think it is “likely” in that range. (We can get more specific on what we mean by “likely” and define, for example, the probability of being in a certain range… but let's save that for a later experiment.)

There are typically two ways you might estimate a measurement uncertainty – from looking at the “spread” of values after making repeated measurements, or from estimating the limit of your precision due to the resolution of your measurement device. Let's look at each in turn.

*Repeating measurements*: Suppose that the quantity you are measuring is difficult to pin down. Maybe it fluctuates with time (e.g. a length that is always varying), you have only one chance to measure it (e.g. the time it takes for a ball to drop), or it is inherently fuzzy (e.g. the diameter of a puffy cotton ball). In this case, if you make multiple measurements, each measurement you make might be slightly different than the last. The best estimate for the uncertainty in a case like this is therefore gotten by looking at the “spread” of values you obtain from repeated measurements.

- If you are able only to make a couple measurements, a simple rule is to estimate the uncertainty as (“maximum value” - “minimum value”)/2. Put into words, you look at the range created by the most extreme values, and chose your uncertainty as
*half*of that. - If you are able to dozens or hundreds of measurements, or if you know something about the statistical distribution of these measurements, then we can use more rigorous methods (like the standard deviation and standard error defined above.)

*Resolution*: The resolution of a measurement device is usually the smallest unit that the device can measure. On a ruler or analog dial, it is the distance between the two smallest tick marks. On a digital device, it is the smallest displayed digit. If you keep finding the same reading on the device every time you make the measurement (assuming you make the measurement the same way), then the “repeating measurement” technique above doesn't help you much. In this case, you can estimate your uncertainty based on how well you can read your scale on a single measurement. There is no fixed rule here; sometimes the uncertainty is equal to the smallest tick mark, whereas sometimes you can say it is closer to one line than the other, so the uncertainty is half a tick mark (or whatever your case may be.)

QUESTION: Now think about the measurements you made today. Which is the appropriatetypeof uncertainty for a single measurement of paper displacement? Which is the appropriatetypeof uncertainty for the average paper displacement?

In addition to measurement uncertainties described above, we also have to be on the lookout for what are called **systematic uncertainties** (or sometimes just **systematic biases**). A systematic uncertainty is an uncertainty in your measurement that is due to some unaccounted for bias in your measurement or your assumptions. It could be that the measurement device you are using is in error (e.g. a thermometer that isn't calibrated correctly or a stopwatch that runs too fast or too slow), that your measurement technique is flawed (e.g. a start or stop signal to a stopwatch always arrives late), or that there is an incorrect assumption about what you are measuring (e.g. measuring the speed of an object that you believe to be at constant velocity, but which is actually accelerating).

Importantly, systematic uncertainties are different from “mistakes” (e.g. writing down a 4 when you meant to write down a 3). Systematic uncertainties are *almost always* present, but they may be difficult to identify or eliminate.

QUESTION: Think about your measurements today. What (if anything) could be biasing your values and how might you come up with a way to test for that? (You do not need to actually perform any additional tests right now. We're just looking for ideas.)

One final comment. We **never** determine an uncertainty by comparing our value to a “known” value. For example, if you measure the acceleration due to gravity to be $g_{\textrm{exp}} = 9.7~\textrm{m/s}^2$, the uncertainty on that value is NOT $\delta g = g-g_{\textrm{exp}} = (9.8 - 9.7)~\textrm{m/s}^2 = 0.1~\textrm{m/s}^2$. Some students may have learned this sort of calculation in high school as the “error” in a measurement, but that is the wrong way to think about uncertainties, and we will develop better ways of comparing two values below.

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 $\mu \pm d\sigma_{\mu}$.

Using the plot with 500 drops, we can now say that the average fall time is $T = 0.997 \pm 0.006$ seconds.

Once you have a value and its uncertainty, how do you report this final result?

This is where the question of significant figures appears. If you do some arithmetic and your calculator spits out an answer of 3.45679213, do *all* those digits matter? Probably not. In order to know how many digits _do _matter, you should look at the size of your uncertainty.

The rules for determining the number of digits to report can be summed up as follows:

- Compute your uncertainty. Keep only one digit in the uncertainty, unless the leading digit is a 1 or a 2.
*Example*: If your uncertainty is 0.543 units, then report the uncertainty as 0.5 units.*Example*: If your uncertainty is 0.0237 units, then report the uncertainty as 0.024 units.

- Look at your value, and truncate your value to the same digit place as the final digit in your uncertainty.
*Example*: If your value is 123.72 units and your uncertainty is 0.5 units, then you should truncate your value to 123.7 units.- Example: If your value is 0.53325 units and your uncertainty is 0.024 units, then you should truncate your value to 0.533 units.

- Put your value and uncertainty together
*Example*: 123.7 ± 0.5 units*Example*: 0.533 ± 0.024 units

QUESTION: Look back at your mean and uncertainty in the mean above for your drops. If needed, rewrite your final measured value and its uncertainty using these significant figure rules.

Below, we provide an online script that can be used to calculate these quantities and to visualize your data. This link uses the Python programming language and runs online through a platform called **Google Colaboratory** (or *Google Colab* for short). You *do not* need to know how to program in Python to use this; consider it to just be a tool (like a fancy calculator).

NOTEBOOK: Complete Part 1 of the program above. Record the values of the mean displacement, standard deviation, and standard deviation of the mean in your notebook.

NOTEBOOK: Save a copy of the scatter plot from the end of Part 1 and add it to your notebook. Are there any outliers that might have affected your results?

Often in experimental physics, we want to compare our number to someone else's.

- Maybe we have a model which makes a prediction for a value. How does our value compare to the prediction?
- Maybe two groups are trying to measure the same thing. Do the two groups agree?
- Maybe we actually expect a result to differ from another known result. How different is different enough?

In order to be quantitative about these sorts of questions, we need to establish some criteria. For this course, we will use a measure called $t^{\prime}$ . (This is related to – but not equal to – something called the *Student's t-test*. If that doesn't mean anything to you, don't worry about it.)

Suppose we have two quantities with uncertainties which we want to compare: $A \pm \delta A$ and $B \pm \delta B$ . I this case, $t'$ is defined as

$t' = \dfrac{A - B}{\sqrt{(\delta A)^2 + (\delta B)^2}}.$ |

In the case where only one value has an uncertainty (for example, when you want to compare a measured value $A \pm \delta A$ to a predicted or literature value $B$ ), this simplifies to

$t' = \dfrac{A - B}{\delta A}.$ |

**Agreement**

If the values are within one uncertainty of each other, it is possible that the difference is due only random chance. We will consider this to constitute agreement: $ |t'| \leq 1$. |

Note that agreement might turn into disagreement if more data is taken and the size of the uncertainties shrink. Remember that we can never *prove* something to be true… we can only say that the current data supports agreement.

**Disagreement**

If the values are more than three uncertainties away from each other, it is statistically unlikely that the difference is due only random chance. We will consider this to constitute disagreement: $ |t'| \geq 3$. |

**Inconclusive**

If the values are between one and three uncertainties of each other, we cannot say with certainty if the difference is random chance or a real disagreement. We will consider this to be inconclusive: $1 < |t'| < 3$. |

NOTEBOOK: Suppose you have two classmates who do this same experiment of dropping squares of paper from 60 cm onto the floor. Angela finds an average displacement of $d_A = 20.7 \pm 1.2$ cm and Mario finds an average displacement of $d_M = 15 \pm 4$. Compare your value to those of your two classmates (and compare your two classmates with each other). Are they in agreement with each other? Are you in agreement with one of them, both of them, or neither? Are they in agreement with each other?

In the second part of this project, you will investigate the relationship between the lateral displacement and the drop height.

The reason that you are 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: $h=v_vt$. 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 di 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 $d_{\textrm{avg}}$ to be proportional to the height.

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

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 $d^2_{\textrm{avg}}\propto t$. 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: $d_{\textrm{avg}}\propto \sqrt{h}$. |

Using your same drop method from Part 1, make measurements over a range of heights. You will need to use your own judgement to determine how many heights to take, and how high or low it is reasonable to go. Compute your mean values, standard deviations, and standard errors.

NOTEBOOK: Remember to record your process and your data in your notebook as you work!

Return to the Google Colab notebook and plot the data yourself (in Part 2).

NOTEBOOK: Include all your figures and final values from the Google Colab notebook. Make sure your notebook includes enough context to understand all that information.

NOTEBOOK: Which model does your data best support? Explain your reasoning. (If you have already learned about the reduced chi-squared parameter in another lab this quarter, you should use those values in your reasoning. If you haven't covered that material yet, use other qualitative or quantitative arguments.)

For this make-up assignment, you do not need to write a summary or conclusion report. Instead, just make sure that you fill out all the sections of the notebook in enough detail and answer all the questions asked.

When you are finished, save the notebook as a PDF and email it to your TA. (Do not submit the notebook on Canvas.)

Your TA will grade the notebook out of 8 points and use this score *in place* of the scores you normally would have received for your missed lab.

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!