Bucket Lab

Background:

We are going to analyze the water draining from a bucket by using Video Analysis by Vernier. Specifically, we will construct a graph of the height of the water vs. time. This is an individual lab. I need every man, woman, and child to do their very own work.

I took a Smart Water bottle, and put dyed water in it. I poked a hole in the side near the bottom, and filmed it with my iPhone (using more compatible encoding) The original video was Ginormous and about 1.2 minutes long, so I used iMovie to make it play 4x as fast, and compressed it to about 15.8 Mb.

Now, the physics of this is that the water velocity coming out of the hole (almost certainly turbulent btw) should be equal to v = sqrt(2gh) where sqrt means square root, g is 9.81, and h is the height of the water above the hole. (This relationship is known as Torricelli's Law) The flowrate therefore is the area of the hole times this velocity. This means that the higher the water, the faster the flowrate, and the lower the water, the smaller the flowrate.

What to do

• 1. Make a hypothesis about what you feel the shape of the graph of water height vs. time will be as the water drains out of the hole. Actually sketch out a graph for me on a piece of paper. Do this before you see the actual data. I will want to see it as a part of your lab.

  2. Gather data from the video at the bottom of this page (Directions at the bottom of the page, <Video>), and put it into a Google Sheet

1. Make a nice scatter graph of "y in cm" vs "time in s" <Video>

2. The curve might be an inverse exponential. Try an exponential curve fit. Does it look right? Try a second order polynomial. Does that look right? Whichever one looks the best, display its equation on the graph. Show the R² value on the graph. The closer it is to unity (1) the better the fit.

3. Click on this link <here> to see the general form of the differential equation that describes the height as a function of time. The input interpretation will give you the equation that I typed in (The rate at which the height decreases (dh/dt) is proportional to (some constant, in this case "k" times) the square root of the height - because of course the flowrate is proportional to the square root of the height...). Scroll down to the "Differential equation solution" and look at the general form. Is it exponential or is it a polynomial?

4. Answer these questions:

   1. Compare the graph to your hypothetical graph you made in step 1 above. Did you guess correctly? How was it different? (if it was)

   2. Why is the graph shaped the way it is? (Why is it steeper at first, and then less steep?)

   3. What fits your data better, the Second Order Polynomial, or the Exponential trendline?

Your lab should include the following: (Make a google doc)

• 1. Your hypothetical graph. (take a picture of it, put it in your doc)

  2. Your nice scatter graph with the model (trendline) you chose

  3. Your answers to the questions in steps 4, 5, and 6 above.

Extra credit

• • Make a graph of |Vy| versus y and do a "Power Series" trendline and display the equation on the chart.

  • How should the speed the meniscus moves down depend on the height of the water? Does this agree with the power of x? Explain what you think the power should be, and what your data says it is.

    • (Hint A₁V₁ = A₂V₂ - where A1 is the area of the hole in the side through which the water issues, A2 is the area of the top surface of the water, and V2 is the velocity you graphed. We know how V1 depends on the height. It's Torricelli's Law.)

Video Analysis Directions (For Chromebook)

• 1. Download the movie file to your Drive by clicking the download link below. Save it to your Google Drive. (should do that automatically on a Chromebook)

  2. Go to this document <Here> and go to the link. (You must be signed into your ttsdstudents.org account)

  3. Click where is says "CHOOSE FILE" and go to where you stored it on your Google Drive and you should see a lovely video of the aforementioned Smart Water Bottle with dyed water in it. (should have used red dye...)

  4. On the left side of the video, click the little "1.0 SCALE" button. It will bring up a tool. Drag the tool so that the black scale line goes from the bottom black line on the red ruler taped to the right side of the water bottle, to the top black line on the ruler. (On my Chromebook this is hard to do because the shaded box covers up the top mark, but if you click back and forth between the "SCALE" and "Add" buttons you can kind of guess and check.) Type that this distance is 20 cm in the two boxes. (20 in the first box, cm in the second)

  5. Advance the video to the last frame, and Move your Origin (The crosshairs thingy) so they are just below the meniscus on the last frame. Click the back button to "Rewind" to the beginning.

  6. Now click on the gear button on the lower right side of the video screen, and set it to advance 10 frames for every data point. (This is so we don't have 600 data points, only about 60.) Since sped up the video by 4x before I uploaded it, we are actually taking a data point every 40/30 ths of a second, or every 1.333 seconds, but that is not important unless you are one of those people obsessed with the truth. Close the settings dialog box.

  7. Now choose the "ADD" button on the top left side. The cursor will become a crosshairs when is is over the video. Make sure the movie is at the beginning, and now we are ready to take data. The app will record in the data table the position x and y, as well as the time for every click you make, and then advance the video for you. It also will place a little dot on the video that becomes less and less helpful as the dots get closer together (i.e. you can't see through the dots - spoiler alert) So what I do is I line up the left side of the crosshairs to where the water level meets the left side of the red ruler. Then the tool makes dots to the right of the ruler.

  8. Put the left side of the crosshairs where the water meets the left side of the red ruler and click. You will see a dot (that doesn't obscure the water level) and the video will advance. Repeat until you have advanced through the entire video. (about 60 dots) You can see the data going into the table on the right, and you can see the graph of your data. Only the blue y values are important.

  9. When you are done clicking, go over to the data table on the right side, and tap the header with two fingers, or do an Alt-tap on the header. (Your Chromebook settings may be different from mine). A menu will pop up, choose "Copy". Go to your Google Drive, and paste the data into a new Google Sheet. (Click in cell A2, and type ctl-V to paste). Label column A "TIme (s)". (Type "Time (s)" into call A1) and column C "y (cm)" and column E "Vy (cm/s)" Delete columns B and D (X and Vx).

How to do all this:

Video Analysis:

Making the Graph in Goooooogle Sheets