# 12.1.1: Testing a Terminal Speed Hypothesis

- Page ID
- 17816

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)## Terminal Speed

### Materials:

- lab sheet and writing utensil
- calculator
- 10 Coffee filters
- step ladder allowing you (or a partner) to reach 2
**m** - ruler with
**cm**units - scale with at least 0.1 gram precision
- spreadsheet and graphing software
- for distance learners, access to online forums, videos, and help features for the spreadsheet software will likely be necessary
- one of the following equipment sets:
- motion sensor + computer with control and analysis software
- video motion analysis app (example)
- camera (slow motion mode preferred) + stopwatch with 0.01

### Observation

We observe that when a body falls through the air it eventually reaches a maximum speed, known as terminal speed, which is roughly 200 **mph**.

### Question

This phenomena raises the question: What determines the value of the terminal speed?

### Search Existing Knowledge

Find an answer for what determines the value of the terminal speed. Write the answer below and also list your source.

### Hypothesis

Our search of existing knowledge told us that one factor affecting terminal speed was the mass of the object.

Provide a qualitative hypothesis on how the terminal speed depends on the mass of an object. That means state if you think the terminal speed will increase or decrease when mass increases. Explain your reasoning.

### Test

To test your hypotheses, without jumping out of airplanes, we measure the terminal speed of coffee filters with varying mass. The terminal speed for coffee filters is much slower than for bodies and they will typically reach terminal speed in less than 2 meters of drop distance. These properties will make our experiment doable in the lab. Your hypothesis was about an object’s terminal speed and mass in general, not about bodies specifically, so a coffee filter experiment will still test your hypothesis.

Measure the mass of the coffee filter and record here:___________

Our method will be to drop coffee filters from a height of at least 2 **m ** and measure the terminal speed. You will need a step ladder and a partner to make the measurements. You can measure the terminal speed using photogates an an acoustic or laser motion sensor if you have access to those in your lab. If not, you can measure the terminal speed by using a video motion analysis app, or by simply filming the last 0.1 **m **(10 **cm) **of the fall with a ruler and a running stopwatch are visible in the video frame.

If using the motion sensor, be sure to only use the section of the speed data after the speed has become constant and before impact. Your instructor will help you find this section of data. Record your terminal speed here:______________

If using the filming method, be sure to film straight on to the ruler, which should be standing up straight on the floor. Read off the time off the stopwatch in the video when the filter passes the 10 **cm** mark and again when it hits the floor. Subtract the first time from the second to find the difference between these times. Divide 0.10 **m **by the time difference to get the terminal speed. Record your terminal speed here:______________

Repeat this experiment for two nested (one inside another) coffee filters. Nesting the coffee filters increases the mass, but doesn’t change the shape of the filters, allowing us to change only one variable at at time. Record your terminal speed for two filters in the chart. Also measure the mass of the two filters and record in the chart as well.

Repeat the experiment until you have measured terminal speed and mass for at least 5 nested coffee filters. Record the number of filters and terminal speed for each in the table below:

Number of Filters | Terminal speed (m/s) | Mass (g) |

1 | ||

2 | ||

3 | ||

4 | ||

5 | ||

6 | ||

7 | ||

8 | ||

9 |

### Analyze

Enter your data into a spreadsheet and create an x-y graph of terminal speed vs. mass. Mass should be on the horizontal (x-axis) because mass is the independent variable (what you are purposefully changing). Speed should be on the vertical axis (y-axis), because speed is the dependent variable (what is changing in response to the independent variable).

Be sure to give your graph a title and label the axes with the variable names and the units of measure.

### Conclusion

Was your qualitative hypothesis supported by the data? Explain.

Be sure to save your spreadsheet and graph. We may use them again during this course.

### Hypothesis Testing Including Uncertainty*

In order to really answer the question about whether or not the experimental results support the hypothesis we need to to think about uncertainty.

Let’s do a little experiment to determine how random error affects the precision of your results. Repeat the final filter set measurement 6 more times and record the results, including the first value you found above, in a chart:

Use your spreadsheet software (or some other method) to take the average and the standard deviation of the seven values. Record both below:

The standard deviation value will serve as an estimate of the precision in our experiment. A new measurement should be within the standard deviation of the average value 68% of the time. We will use the precision provided by the standard deviation as our estimate of the uncertainty in our final measurement. Ideally we would base our average and standard deviation on more than seven values, we will use only seven in this learning situation, simply for the sake of time.

Add error bars to the terminal speed data in your graph, setting the size equal to the standard deviation you calculated.

Considering the error bars, does the data support your qualitative hypothesis? Explain your reasoning.

So far we have ignored systematic error. Systematic errors can be difficult to recognize and even more difficult to quantify. We must always be on the look out for sources of systematic error. Can you provide a possible source of systematic error in your experiment? Explain.