Physics of Bandaloop Dancer

The other day, I was shown a video of one of the most exciting looking things I've seen in some time:

 

The group of dancers is called Bandaloop and among other things, they dance on the sides of buildings and cliffs by being attached via rappelling ropes. The long ropes keep them from falling, and allow them to make incredible leaps with long hang times.  A video interviewing the founder says some days, from tall buildings, she can get jumps with hangtimes of 9 seconds or more. 

My first reaction to the video was "WOW! I wanna do that!".  My second reaction was "What's the physics behind that?  Nine second hangtimes? Really?!"  

Since I can't try that anytime too soon, I must instead attempt to describe the physics behind them. Here are the pictures I drew on the side of the napkin:

I didn't have lots of data to go on, so I initially estimated that the Length of the ropes (L) might be approximately 30m, and if harnessed at the center of the body, that puts x at approximately 1m.  At best jump, I estimated approximately 5m out from the building. Since the sin of the angle θ is x/L, the inverse sine of 1/30 and 5/30 suggests that θ ranges between 2 to 10 degrees. At angles this small, sin(θ) and tan(θ) are nearly identical, suggesting that y and L are nearly identical too, and I'll be interchanging them occasionally. This is not true as the rope gets smaller -- so relatively tall buildings and tall cliffs are important.

Next I drew a free body diagram of the forces acting on the dancers as they are away from the building.  There are primarily two forces acting on the dancer -- Weight pulling the dancer down, and tension in the ropes pulling the dancer at angle θ up and in toward the building.  That angled force I broke into components Tx acting in towards the building, and Ty acting to counteract the dancers weight.  Because the ropes are so long, the height of the dancer doesn't change significantly, and since the dancer isn't really moving much vertically, we can say that the forces are balanced vertically.  That is, W = Ty. Since W = mg, this means Ty = mg.

Horizontally, the forces aren't balanced, and so whenever the dancer is in the air, there is a portion of the tension Tx which acts to pull the dancer back in toward the building. This unbalanced force is a net force, and so we can write another equation: Fnet = Tx. Since Fnet = ma, this means Tx = ma.

Finally, Tx and Ty are related to the angle by the tangent relationship, such that tan(θ) = Tx/Ty, and after multiplying, Tx = Ty*tan(θ).

After a few substitutions and a little division, we find a formula for the acceleration inwards toward the building that the dancers feel:

It's surprisingly simple and clean -- the acceleration the dancers feel is just a multiple of gravity. Since θ ranges from 2 to 10 degrees, the dancers feel acceleration toward the building ranging between approximately 1/30th to 1/6th that of normal gravity.  For comparisons sake, the acceleration of gravity on the moon is about a 1/6th of that on earth, which according to my rough estimates is about the most that the dancers would feel inward toward the building on their most extreme jumps. Most of the time they are just a meter or two away from the the wall they are feeling much lower attraction toward earth.  

Put another way, the dancers feel approximately 1/30th of their "weight" inward toward the wall, so a 120lb dancer might feel only 4 lbs of forces inward. Imagine how easy it would be to jump if you only weighed 4 lbs, but had the strength of a professional dancer!

Which suggests a different approach to this problem. Let's figure out how far out from the building a dancer ought to be able to get, assuming they can jump "off the wall" with as much speed as they can normally jump off the ground.... Assuming that a person can normally jump to a height of 0.5 meters under typical gravitational acceleration of -9.8 m/s/s, we can use the "no time" formula:

They would have to jump at a velocity of approximately 3 m/s, yielding a hang time of approximately:

 

Leaping with the same initial speed (3 m/s) off the side of the building with an acceleration of 1/6th of gravity as these dancers feel would allow them hang times of 6 times as much:

  

Not quite the 9 seconds claimed in the video, but more on that later.  Substituting half of this time (because the maximum height occurs halfway into the trip) into the kinematics equation allows us to calculate the maximum "height" off the buildings this dancer could reach:

That's pretty close to the 5m I estimated from the video.

Now the hangtime of 3.7 seconds is far less then the the claimed 9 seconds of hangtime, and even my closer observation of the video suggests a few jumps were more than 5 seconds long. One way to get more hangtime is to jump with more speed - something that's quite possible with stronger and trained legs. Remember I started with someone able to jump 0.5 meters -- and I'm sure a strong dancer could leap higher. 

Another reason the kinematics equations don't give us enough hangtime is that the acceleration is not constantly 1/6th that of gravity -- often it's way less than that even! Smaller accelerations, like those felt close to the building when θ is small, would increase hangtime significantly. Unfortunately, I've forgotten the formulas for how to deal with accelerations that aren't constant -- although I'm sure a few google searches could refresh me. 

Either way, the Bandaloopers certainly can experience tremendous "jumps" due to the low horizontal forces they have to fight on their rotated worlds, and simple first-year physics concepts help reveal why.  

Now, perhaps someday I'll be able to actually try it for myself.   

Conjunction Junction - Wait, what's conjunction?

The other night as we drove home late from softball, my wife started asking me a few questions about astronomy terms, and so I thought I might write down a few of them.

• Conjunction: A conjunction is when two (or rarely three or more) objects are close together in the sky.  This is typically spoken of in terms of the planets. For instance, just this week Mercury and Venus experienced conjunction, and were very close in the sky.  With so many moving objects in the sky, a conjunction of some sort occurs just about every month -- and certainly so if you include the moon as one of the objects 

Jupiter and it's moons
 "being occulted"?
(I don't think that's proper English)

• Occultation: A very special conjunction where objects appear so close together in the sky that one object actually passes behind another. The most common types are when the moon passes in front of some object, and for a short time, that object is hidden behind it. I have never had a chance to observe this. Here is a list of lunar occultations for this year and you can see that the moon passes in front of stars all the time (nearly every day) but in front of major planets only a few times. And during those times, you can only see the occultations from generally small locations on earth. 

• Transit: A transit is another type of conjunction, when a smaller object moves in front of another bigger object. The most famous transit is when Venus transits the sun, an event that occurs twice every 120 years or so. The most recent was June 5, 2012, so I'm sorry -- if you didn't see it then, you probably won't see it ever.  I made sure to watch it, and took this picture. The transits of Venus in 1639, 1761 and 1769 are of historical interest, because they helped scientists get an accurate measure of the distance from the earth to the sun.

• Syzygy: This is just too cool of a word to leave out, even though I've never seen it written anywhere except in a glossary of astronomical terms. It's a great Scrabble word, worth 25 points, for those rare (impossible actually) occasions when you have 3 y's.  Essentially, a syzygy is whenever three astronomical objects are all in a line.

• Eclipse: An eclipse is when the sun, moon, and earth are in syzygy -- and depending on the order of the three, and when the syzygy occurs, you might experience an eclipse.  Every 14.5 days the three are aligned in some way, but most of the time the moon is slightly above or below perfect alignment, and so only a handful of times each year does some kind of an eclipse occur.  Here's the next ten years of eclipses.  The next total solar eclipse that will be fully visible in the North America will be on August 21, 2017, an event I'm planning on driving down to see. 

• Opposition: When a planet is at opposition, that means it is on the opposite side of the earth from sun. This is for the planets further away from the sun then earth -- and is usually the best time to observe them.  The planet is usually the brightest then -- and highest in the sky (along the local meridian) at midnight.
  
• Elongation: Elongation occurs for inferior planets (Venus and Mercury) and is when appears the farthest away from the sun. This marks the best times to observe Venus and Mercury -- when they are their brightest, and furthest away from the sun's blinding glare. 

Reflections/Refractions on a rainbow:

Today we had a thunderstorm roll through an hour before sunset, and afterward a beautiful, full, double rainbow appeared.  It was so beautiful that I woke my daughter Abby up (actually, she hadn't quite fallen asleep yet) to take her outside and see it.  After about one minute Abby became more fascinated in the neighbors dog, but still, she said it was pretty.

Afterward, I came in and by the time I had posted these pictures to facebook, everyone else had already posted pictures to facebook about the same rainbow.  It reminded me of the second to last page of my difficult physics exam this semester, which was all themed about different types of severe weather, such as the energy of a falling hailstone to the current in a lightning bolt and the centripetal force of cows stuck in a tornado.  At the end of the exam, I posted this page, reflecting on the promise God made to never again destroy the world with a flood.  In it, I describe how each one of us sees our own individual rainbow - a testament to the way that God gifts each one of us separately and uniquely.

A rainbow is God's sign that he provided to us as a reminder of his covenant to never again destroy all life with a flood. Though God is the speaker in Genesis 9:16, I like to claim the verse for myself that says: "Whenever the rainbow appears in the clouds, I will see it and remember the everlasting covenant between God and all living creatures of every kind on the earth." It reminds me that though the storms of life will come, my God will not let me go.

A rainbow is a consequence of light from the sun being refracted as it enters a water droplet, reflected off the back of the drop, and refracted again as it leaves. This refraction causes the light to split into different colors. Each droplet sends a specific wavelength of light back to your eyes, which you interpret as a specific color. The entire collection of water droplets in the sky, all producing different colors -- or, if you will, all singing different notes -- produces the symphony of light that you enjoy.

Even more amazing is the fact that the person right next you is experiencing their own unique rainbow, as the rays of light necessarily must travel at different angles to reach their eyes. Therefore, a droplet that you see as red, might be producing yellow for your neighbor, and a droplet they see as violet, you might not even see at all. 

Reflect (No pun intended. Ok, maybe a little.) on these thoughts for a moment, and then proceed to the final page to share some of your own thoughts from the entire year. If time allows, feel free to additionally share some of your own thoughts on rainbows on this page.

Are you smarter than a calculator?

Lately, I have been noticing how dumb our calculators are. It's become kind of a running theme in my classes, where I've been teaching how to use different graphing tools, and I say several times a week "and remember, you have to be smarter than your calculator" or "you have to help your calculator..."

The calculator doesn't think like humans. God created us--not calculators--in his image, and I believe one aspect of that is the ability we have to reason, to notice patterns, to create, to organize, to see. These are all things that calculators, and in general computers or machines, are all pretty bad at. They are improving, because our minds are helping to generate better and better machinery, but they still don't work like humans.

Perhaps the best example of this is the Captcha messages at the bottom of so many websites. Computers and bots are still horrible at "seeing" things. Most humans can interpret those pictures and type letters or numbers properly, but that relatively simple operation is difficult for most computers. That's because we think about the problems entirely differently.

Likewise, our calculators think about calculations entirely differently than us. We can think algebraically and manipulate symbols, variables, and even numbers in symbolic ways that allows us to simplify problems, or calculate values exactly. Most calculators don't think in that way at all, but are programmed with different algorithms that work with really precise approximations and quick calculations. Even the slowest earliest calculators can do this sort of thing faster than all but the freakest of humans -- but I haven't seen any calculators that are good at playing What's the Word.

Here's a bullet list of items that I've noticed lately:

• In PreCalc we've been studying complex numbers. One assignment the other day was to calculate i¹⁷ which is easy to calculate by recognizing a pattern.  i, i⁵, i⁹, i¹³, and i¹⁷ are all equal to the purely imaginary number i, but the calculator spit out -1E-13 + i.  Yes, the -1E-13 is a ridiculously small number, close to zero, but it shouldn't be there AT ALL! What strange algorithm does the calculator use to calculate that instead of just recognizing the pattern like humans?
• Similarly, some versions of the calculator were not able to convert some of our operations involving complex numbers into exact fraction form -- where as we could. Some calculated approximations (admittedly better approximations than we could find in anything short of five/ten minutes) but several others gave an ERR: data type message instead
• In Algebra 2, we gave been using the calculator to calculate summations, and the notation for summations is sum(seq(function,VAR,start,end)) and we have been laughing at the fact that even though our functions only have one letter in them, we still need to write that variable again. I understand you could certainly have many variables in a function and then you'd have to specify which one is the index -- but when there's only one, you'd think the calculator would be able to figure that out.
• In Algebra 1 we've been graphing systems of equations, and numerous stupid calculator quirks have popped up. Though we set the word problems up with sensible variables like N for the number of nickels and D for the number of dimes, when we went to graph things, we had to use the letters X and Y. Again, you could maybe give your calculator the benefit of the doubt because maybe those letters are going to be used for constants (like I do in physics storing 6.67E-11 in for G) but...
• Then we try to calculate the intersection of two lines and we have to tell it which lines we're interested in and help guide it towards the solution. Seriously?! There's only two lines on the screen! And they're lines! Not curves!  
• If the intersection isn't on the visible window screen, the calculator won't be able to find it for you -- you need to realize that those lines will intersect above, left, right, etc. of the screen and adjust the window yourselves.
• And what's with providing the answer as 1.999946 when it's clearly and exactly 2?  The algorithm that calculates the intersection necessarily has limits to its precision, and sometimes those fall short. My students better not ever report an answer of x=1.999946 to me.
• To be fair, let's pick on non-TI84 calculators -- one of my newfound favorites is the app MyScript Calculator which interprets my handwriting and calculates things for me.  I'll admit, I played with it for a good 30 minutes after downloading it -- only true math nerds play with their calculators right?  But I noticed pretty quickly that it's trig values didn't always calculate properly, which was a bug that their updated version supposedly has fixed. I knew that because I knew the limits of sine and cosine values, and even had several memorized -- and also can estimate relatively well and had ideas of what the answers should be ahead of time.

Greek Letter Shortcuts in Microsoft Word

When typing up notes, worksheets, and assignments, I often find myself needing to type some special mathematical symbols, such as π. Other Greek letters are useful too, such as Δ for change in. The lazy thing to do is just type pi, or delta, but there are a handful of ways to insert these symbols into a document.

Character Map

• In Windows, you can find the old Character Map -- a program I've seen since I used Windows 3.1 growing up. You can find it by pressing the Windows button and searching for character map -- and then the window on the right will pop up. In it you can find a wealth of characters that you can copy onto the clipboard and paste into whatever program you're typing in. I used it to type the π and Δ symbols above.  Though this technique can work, and work for multiple programs, I don't use it often
  
  
• In Microsoft Word and PowerPoint, you can find Insert Symbol in the ribbon along the top. Like Character Map, it gives you a list of hundreds of different symbols that you can search through to use. The window looks like this:

• If you notice the bottom of the box there is a shortcut key programmed in for the letter π. In Microsoft Word, you can use this code to enter in a π symbol without using this window -- much faster!  To use the code, type 03c0 (those are zeroes) and then hold Alt and press X.  You'll see that the 03c0 becomes a π symbol! Magic! And many other characters have a character code similar to π. My problem is that there is no way that I'll remember each of those strange character codes every two weeks or so that I need them. So I would have to look them up online and in the time it takes to do that, I could have just found it in the insert symbol box.
  
• A related technique is to use an Alt code.  Hold Alt and press 227 on number pad, then let go of Alt and you'll see a π symbol appear.  Many other letters have a Alt code, but again -- how am I supposed to remember which codes are which?  And why in the world is the Alt code for π 227 and not something reasonable like 314?!
  
• My favorite solution to this in Microsoft Word is to use the shortcut button to create my own keyboard shortcuts for each of the symbols I use often!  If you click on the button "Shortcut Key" it will bring up the following form:
  As you can see, I have created a keyboard shortcut that I'll remember, and stored it in Word so that when I return a few weeks later, I can find the π symbol more quickly!  For me, it made sense to type Alt G, P.  My line of reasoning is that it's a greek letter, Pi.  I have gone ahead and assigned shortcuts similar to this for all the Greek letters that I use on a regular basis:
• Alt G, D for lowercase delta
• Alt G, Shift-D for uppercase delta (for "Change in")
• Alt G, A for alpha (for angles)
• Alt G, B for beta (for angles)
• Alt G, Q for Theta (for angles)
• Alt G, T for Tau (torque in Physics)
• Alt G, r for Rho (density in Physics)
• Alt G, W for Omega (rotational velocity in Physics)
• Alt G Shift-W for captial Omega (Ohm symbol for resistance in Physics)
• Alt M, . for the mathematical symbol for multiplication
• Alt M, 2 for the squared symbol
• Alt M, 3 for the cubed symbol
• Alt M, R for the square root symbol 
• While on the topic, I should also mention auto-correct as an option. I remember using that quite frequently during my Spanish classes.  I made it so typing n~ and a' and such created the accents and symbols -- though if I used those characters much now I would probably create a series of Alt codes for them, perhaps with Alt S, N and Alt S, A, etc. 

Reflections on J-Term 2013

We just finished J-Term and I have so much to write about, but so little time to write, so let me just do quick summaries and if I have any time (this summer?) I'll come back and elaborate.

For those of you who don't know, at our school J-Term is a week-long opportunity for our students to take some unique classes and learn things their teachers don't normally get to do during the school year. For instance, many students took iPad video making, or an interesting Hunger Games exploration, knitting, chess class, etc. The teachers suggest offerings and the students sign up for three different classes they'd like to take.

This year I offered two classes: Astronomy (which I have taught before) and a new class which I called "Did You Get My Email?" but might more formally be called Digital Communications.

Star and Planet Locator
by Edmund Scientific

In Astronomy we learned a 15-20 constellations, discussed how to use a Planisphere, the idea of altitude and azimuth, how to find the planets along the ecliptic, and how the sun moves through different constellations (the zodiac) throughout the year. Next on the list would have been declination and right ascension, but we ran out of time.

The Star and Planet Locator made by Edmund Scientific is an great tool for teaching these concepts -- and at only $3.95 per unit it's one of the cheapest I could find.  I bought mine a few years ago and kind of remember a 25 for $50 deal so if you're interested in a classroom set, look around.

These worksheets I offered:

• Learning the Constellations: Identify the constellations you see in different maps
• Using a Star Finder:
• Using a Star Finder Part 2:
• Seasons: Questions regarding seasons and sun on star finder
• Angles Between Stars: measuring altitude and azimuth with hands, degrees, etc.
• Estimating Location of Stars: practicing altitude and azimuth with a star finder
• Celestial Coordinates: practice with declination and right ascension
• Stars and Heavens in Scripture: Bible passages referring to the stars and heavens
• The Star Clock: Using the big dipper to tell what time it is

The other class I taught was new to me -- Digital Communications. I'll admit I'm not proud of how this class turned out because I didn't put the time into it over Christmas break that I should have. In this class we learned about a ton of different technologies, leaning quite heavily on "How Stuff Works" descriptions of: the telegram, telephone, television, computers, hard drive, cd player, text messaging, email, radio, etc. We also studied binary numbers, and spent some time describing how computers convert all information into numbers, which are all converted into binary, which can mean everything can be stored ultimately as a handful of 1's and 0's somewhere.

I also did some hands on materials, though I had ambitions of doing way more. We played around with simple circuits, hooking up batteries and lights. We made a few electromagnets, and I showed them a homemade "byte" -- 8 bits -- which I made with just a piece of wood, 8 nails, and a about 40 ft of wire. I never used it in anyway besides holding it up occasionally when we discussed that 8-bits define a character in Ascii, or that three of these 8-bits define a color of an individual pixel in a picture, and so on.

I learned from this that I enjoy doing things hands-on and should take more time to make that happen in my classroom. I learned that radio shack has a lot of small circuit components for sale, such as LED's, solar panels, resistors, switches, etc and I have a lot of material now that I'll be able to use in our electricity unit in physics.  And I learned that classes will survive, even if you are fully prepped for them. Maybe that wasn't the lesson I should have learned -- but I did.

The Constellation Gemini

This post is the one of a series on constellations and posted throughout the year as each constellation comes into prominence.

Gemini is an important constellation to learn for a number of reasons. Known as the twins, it is home to two of the brightest stars in the sky -- Pollux and Castor. Pollux on the left is the 12th and Castor the 16th brightest stars in the Northern hemisphere. These stars are the heads of two twin brothers standing side by side. On a good night, you can see lines of stars outlining the bodies, as shown below:

Gemini
Image from Wikipedia

Gemini is one of the 12 zodiacal constellations, which means it lies along what's called the ecliptic. The ecliptic is a circle of constellations surrounding us which lies along the same plane as the planets and the solar system as a whole. Therefore, any planets you see are always somewhere along this plane. Gemini is therefore useful to know as planets are often located nearby.  For instance, the bright planet Jupiter will be approaching Gemini early 2013 and spend late 2013 and most of 2014 within the region.

Being along the ecliptic means that the moon will pass through the constellation once a month.

Being along the ecliptic also means that the sun will pass through the constellation (rendering it invisible of the brightness of the sun) once a year. The sun passes through Gemini from May 21 to June 20 each year. During the winter months, the sun is in the opposite side of the sky which makes Gemini an easy constellation to see.

Below is a map of the surrounding constellations for the evening hours in December.

2012 December skies ~ 7pm
2013 January skies ~ 6pm
Image by Skymaps

As you can see from this image, Gemini is pretty easy to spot -- being above and to the left of the familiar constellation Orion. Finding the bright head stars should be easy to do, but next time you have a clear night, see if you can identify some of the other stars making up the body. I have often seen them as forming the Greek letter Ω (Omega).

Of course, a description of Gemini would be incomplete without mentioning the Geminids, a meteor shower that is one of the best in the year. During the nights around December 13 and 14 each year, many bright meteors (sometimes around 100 per hour!) can be visible. They emanate from Gemini - which means they will go outwards away from Gemini. They don't always begin in that constellation, but their paths will tend to point away from Gemini more often than not. Most meteor showers are best observed in the wee morning hours.

A good binoculars object, M35 is a nice cluster of stars that will be my goal to find in the next month. It has the width of the moon, and contains well over 200 stars, but most are invisible to the naked eye. Castor is supposedly juggling it on his foot like a soccer ball.

A Ton Of Snow

The other day my students were taking their pre-calc exam and due to my writing it too long, many needed to stay after class was over to finish. So many in fact, that I emailed "Wow, I have a ton of students needing to stay after!" to my principal.

After hitting send, I was instantly annoyed that I had used "ton" as an exaggeration -- something I have been known to chide my students about. But I was comforted when I did a quick estimation and realized that I was actually fairly accurate:

As it turned out, I had 14 students in my room, and many of them were the bigger athletes (certainly bigger than 120 lb) so I'm sticking to my original statement as literally true. And is 120 lb an appropriate average weight of a student?  Who knows.

Well, today I was out shoveling snow for the first time this season, and I caught myself again saying "This is a ton of snow!"  Well, was it? Was it really?

To calculate the weight of snow, I need to find the volume of snow on my driveway, and multiply it by the density of snow.  The volume of snow is easy enough to approximate -- I'll assume I have an even rectangular driveway with the same height of snow everywhere, and so volume of this rectangular prism is just length * width * height:

 The density of snow is a little more difficult, as there is heavy snow, light fluffy snow, solid ice, etc.  Wikipedia says snow has a density of anywhere from 8% - 50% of water, depending on many things, but mostly on how compacted it is as it melts, freezes, melts, refreezes and more snow falls on top of it. This was relatively new snow, and so I'll assume it has a density of about 25% of water.

The density of water is 1 kg/L.  This "coincidence" of having such a clean number is actually not a coincidence at all, but was by design -- as 1 kg was originally defined to the weight of 1 liter of water as the metric system was being invented. Since then we have defined the kg more precisely than that using more complicated methods. Though it's no longer exactly 1, this value remains accurate enough for our purposes (I think it's 1.003 or something close?)  Converting kg to pounds and L to cubic inches is a tough exercise:

If my snow was 25% the density of water, than:

The weight of the snow on my driveway is then:

Looks like I wasn't exaggerating after all! I moved over a ton of snow today!

Answer Key's to Worksheets

I believe that students do need to practice a skill in order to get good at something. This often translates into lots of questions and problems and opportunities for students to practice. Typically, for my math and physics classes, this takes the form of worksheets or practice problems out of the book. 

One thing I try to make sure I have for my worksheets or assignments however is some way for the students to know if they are getting it right. 

For my algebra assignments out of the book -- I remind my students that math textbooks almost always have an answer section in the back, and I tell them to check their work as they go. (One thing I have not done that just came to mind is model to them what that looks like in practice, and how to deal with mistakes...) For assignments that do not have answers, I have been known to write all the answers on the board, or on the bottom of a worksheet, but in random order. This way the students have to come up with an answer, but if they see that answer on the board then they have some confidence that they did it right. To avoid process of elimination at the end, I usually throw a few more answers up than I assigned.

My physics assignments typically come from several sources -- a hardcopy textbook, made up questions on worksheets, or several handouts of supplementary practice problems. Most of these do not have a ready made answer sheet, or have multiple parts that should be checked along the way before arriving at the "final answer". For many of these assignments, I provide the answers in a multiple choice format:

      m = {1.20,     1.47,      1.93,       2.31,     2.88,    3.06} kg

One of the six options listed is the correct mass, and the other 5 are just random distractors. The idea came to me when one of the students complained to me "Can you just give us the answers?".  At first I thought, "Of course not!" but then I remembered my assignments at MSU where we had to enter our answers in online, and they would tell us if we were right or not. I remember being frustrated at these assignments, but in a good way because I kept returning to the problem and trying to figure out what was wrong until finally I found an answer that worked. I haven't taken the time to find a way to do that with my assignments yet, but having the answers available for students is a step in that direction perhaps.  

To help me come up with such answers, I set up an excel file that I can enter the correct answer in one cell, and it will come up with six answers that are all reasonably close in size compared to that answer.  The correct answer is randomly placed somewhere in those numbers:

If you want to try it for yourself, you may download it here.  To adjust where the answers round to, select the options and push the rounding button in the number tab above, highlighted in yellow above. 

To create a worksheet for the students then, I typically have this file open and copy and paste the answers in.  One thing I have found necessary is to "Special Paste" them in -- by pressing Alt E, S and paste them as unformatted text instead of cells from a spreadsheet. That way the formatting looks good, and the numbers in the worksheet are fixed instead of linked to the constantly changing excel file.  

The Coathanger

The last couple of nights have had some clear skies in Michigan, and since I can't sleep (thinking about the imminent second daughter coming any day now) I have been doing a lot of stargazing lately. I was reminded of the amazing creativity and power of God as I brought out a new simple tool to help me stargaze, a pair of binoculars.

The first time I saw in an astronomy book a picture of a family using binoculars to look at the stars, I thought "Wow -- how desperate!"  I had major skepticism on how well they could really help the view. But as I tried them this week (and my binoculars are by no means that spectacular -- they're small) I must admit being amazed at how much more I could see. Yes, it made things slightly bigger -- mine were 7x -- but more impressive to me was that you can see more stars!

The Coathanger

Nothing made that more evident then when I discovered a new constellation (new to me that is) called the Coathanger. I was just wandering around looking at the stars (actually, following one of satellites that I couldn't see with the naked eye) when I ran into a tight grouping of stars, technically an "open cluster" that I instantly recognized as "The Coathanger" that I had coincidentally read about that morning. Typically constellations don't look anything like they're supposed to for me, but this one most definitely did.

Technically, the recognizable shape of stars called the Coathanger is an asterism, as opposed to a constellation which is really just a chunk of the sky. As another example, the big dipper is another asterism, that is located in the constellation Ursa Major.  What made the Coathanger so impressive to me, was the fact that I cannot see it at all, without my binoculars. Click here for more information on Brocchi's Cluster, or Collinder 399, which is the catalog name for the cluster containing the asterism of the Coathanger.



If you'd like to find the Coathanger yourself, Here's a map of the stars that are visible in the evenings of midsummer in the Northern Hemisphere. If you need to look up different latitudes or months, try this more general link. 

Before we find the Coathanger, let me set the stage for you. If you have the map in front of you, it might help as I describe what you can see. To the northwest you should see the Big Dipper. I recommend you start by looking at the Big Dipper through your binoculars, and practice moving from star to star to get a feel for how big the binoculars are, and how many more stars you can see than you're familiar with.  For my set, each major star in the 7-8 that make up the dipper required me to move about one field of view in my binoculars, and it took some time before I could confidently move from star to star, so don't be surprised if it's a little difficult at first.

To the right of the big dipper, in the northeast, you'll find the "W" which is in the constellation Cassiopeia.

Behind you, in the south along the horizon, you should be able to find Scorpius, which is in my opinion a fairly obvious constellation that looks like a scorpion. Just behind it, to the left, you might be able to spot "the teapot" in Sagittarius.

Sky Map provided by skymaps.com
Looking east, this snippet covers from
about 45 degrees up to directly overhead.

Now turn towards the east, and then look straight above you. The point directly above you is called the zenith and the bright star Vega is pretty close to the zenith during the summer evening hours. You should come back and look at the stars around Vega in Lyra, as there are some beautiful and obvious double stars there that pop out with a binoculars, but try to locate the other constellations as shown in the snippet of the map above.  Below and to the left you should see what I call the "Northern Cross" which is in the constellation Cygnus, a swan flying south over the Milkyway river.  Below and to the right is a less obvious constellation of stars called Aquila, an eagle who is also flying south.

I find the Coathanger most easily by finding the three right stars in Aquila's tail, (the middle one is the brightest, and is named Altair) and following that line up about two binoculars widths (10 or so degrees). You should be able to see CR399 labeled on the portion of the map above. CR 399 is actually a little wider than a full binocular width for me, and coat hanger is slightly left of the line. If you see it, you'll know, because the line of stars marking the hanger part is so perfectly straight that it jumps right out at you.

Let me know if you spot it!  Also, be sure to say high to Jolly Mon and the Dolphin, my favorite constellation (Delphinus) while your looking in his neighborhood, and have fun checking out the skies!

Metric System Prefix Fufu Song

Little basic FuFu, hopping through the forest
Scooping up the millipedes and boppin' 'em on the head
(spoken) Down came the 10-to-the-third fairy, and she said:
  "Little basic FuFu, I don't want to see you,
   scooping up the millipedes and boppin' 'em on the head.
   ... (spoken) Now I shall decrease each of your three dimensions by a factor of ten: poof!


Little milli-FuFu, hopping through the forest
Scooping up the micropedes and boppin' 'em on the head
(spoken) Down came the 10-to-the-third fairy, and she said:
  "Little milli-FuFu, I don't want to see you,
   scooping up the micropedes and boppin' 'em on the head.
   ... (spoken) Now I shall decrease each of your three dimensions by a factor of ten: poof!

Little micro-FuFu, hopping through the forest
Scooping up the nanopedes and boppin' 'em on the head
(spoken) Down came the 10-to-the-third fairy, and she said:
  "Little micro-FuFu, I don't want to see you,
   scooping up the nanopedes and boppin' 'em on the head.
   ... (spoken) Now I shall decrease each of your three dimensions by a factor of ten: poof!

Repeat with, nano,pico/femto/atto/zepto/yocto.

  ... (spoken) I suppose you'll never learn, will you?  Back to normal!

Little basic Fufu, running through the forest

Hiding from the kilo-birds who bop him on the head

(spoken) Down came the 10-to-the-third fairy and she said,

   "Little basic Fufu, how I hate to see you,

    Hiding from the kilo-birds who bop you on the head"

    (spoken) Lets try increasing each of your three dimensions by a factor of 10: poof!

Little kilo-Fufu, running through the forest

Hiding from the mega-birds who bop him on the head

(spoken) Down came the 10-to-the-third fairy and she said,

   "Little kilo Fufu, how I hate to see you,

    Hiding from the mega-birds who bop you on the head"

   (spoken) Lets try increasing each of your three dimensions by a factor of 10: poof!

   Repeat with Mega/Giga/Tera/Peta/Exa/Zetta/Yotta

Some Systems Videos

Hey guys,

It's been a while since I've posted anything since I've been working especially hard creating a lot of videos for my classes at school.  I'm now back into a flipped unit in physics, studying Torque and rotation with my students. If you're curious, you can follow the unit at our class website. I am creating this unit as a sample of a flipped unit for my Masters project at Cornerstone University, so I'm taking more time perfecting it than I normally might -- thinking about tying all the things I've learned together.

I've also recently created several videos for my algebra 1 students.  Though I am not flipping that class -- I do have a lot of students who are absent a lot, and I thought it might be worth it for those students for me to post videos of the lectures online.  Since I already have all my equipment up and out and ready for my physics classes, I talked through my algebra lessons too and here are those videos:

This is an introduction to systems of equations, and my explanation of the graphing method of solving systems:

z

Here's the substitution method, which we'll be studying next week -- I like the colors in this video -- it turned out pretty:

And here's the elimination method, or what my text book calls "linear combinations".  Elimination is a much sweeter name.

This is probably one of my favorite lessons to teach -- I just think the combining equations and one variable disappearing is a beautiful magic trick -- makes me feel like a mathmagician, as some of my students say.  The only thing cooler than adding two equations together and having variables cancel out, is dividing two equations together and having variables cancel out, a trick I often do in physics, especially when sine and cosine values are around.

Crash Test Data

We've been spending the last couple days in physics analyzing the following video:

We used Tracker Video Analysis to analyze the force the dummy felt without a seatbelt, and compare that to the force the dummy felt with a seatbelt and airbag combination. Since we have been studying momentum, I was hoping that we could find that though the dummy experiences roughly the same impulse (change in momentum) either with or without a seatbelt, the dummy experiences greater force without a seatbelt due to the impulse occuring in a shorter interval of time.  The formula for impulse is after all Δp = FΔt which yields F = Δp/Δt which is bigger when Δt is smaller.  Initial findings from the class have not been conclusive whether Δt is indeed smaller, but since the students are still writing their labs on the subject, I won't elaborate here.

I will however post the graphs I found.
First the boring data -- the car - a position versus time graph (inches and seconds are units)

 and a velocity versus time graph (in/s and seconds are units).

Now for the dummy without a seatbelt - a position versus time graph (inches and seconds)

 and a velocity versus time graph:

Finally the dummy with the seatbelt -- looking at just the initial collision (inches and seconds)

 And velocity versus time (in/sec and sec)

Since the close up view of the dummy did not show the whole picture, I did another track of the whole dummy with the seatbelt on, this shows the initial impact with the airbag, but then the subsequent hitting against the chair and finally coming to a stop. Again, units are inches and seconds

 and for the velocity graph: in/sec and seconds.

If the Δt does not prove conclusively that Force is diminished, then I may need to give a quick primer on pressure -- as I'm sure the pressure on the dummies forehead in the crash without a seatbelt (notice the glass shattering!!) is much higher than the pressure the dummy felt smothering his whole face (notice the paint left behind) on the airbag.  Otherwise my students might wrongly conclude that wearing their safety belt is worse than riding without.