Sunday, March 18, 2012

Truth in Advertising!

I saw this commercial tonight during one of the NCAA tournament games (these are basically the only 4 days of basketball that I watch all year.  UPDATE: viewing extended at least one more day: Go Wolfpack!).

It's all here: assumptions, initial values, what seems like a motion map... let's dig in!

The captured information:
This seems like it's intended to be the initial velocity, but the ball in the frame doesn't seem to be moving yet.  Known so far:
  • Ball mass (incorrectly labeled weight): .624 kg
  • Ball diameter: 23.8 cm (I'm not sure where this value comes from, since it actually gives a circumference that's a shade under what Wikipedia tells me is the minimum allowable under the rules)
  • Initial (?) velocity: 9 meters per second
  • Given that we know where the ball's going to land, they tell us that we're now at 0 of 24 meters or (presumably) horizontal distance.
This distance is a little shady to begin with.  I find that the legal length for the court is 94 feet, and that the free throw line is about 19 feet off of the baseline. 24 meters puts us a yard or so inside the FT line, but Laettner's definitely on the near side of it, and he's reaching towards Hill, shortening the distance further.  Perhaps this was an older standard - can someone tell me if the courts are longer now?  Are they looking at the distance that the ball would've gone, had it hit the ground?

It's interesting that, using that original video and scaling the court to the foul line distance, I get just about the same distance (23.4 m vs. 24 m) that they got, without including the small bit of added distance from the fact that Hill as a little further down the baseline than Laettner was when he received the ball.  It doesn't seem like this distance can be correct, given the court dimensions (the angle between the two would need to be about 22 degrees, putting Hill 9 meters down the baseline from the hoop, which isn't reasonable).  Perhaps we both made the same video analysis error - checking against the court dimensions makes me think that's the issue.

The back-of-the envelope calculation for the flight is interesting.  The time stops a few times during the video, but going back to original footage, I used Tracker to get a flight time of 2.069 seconds.  For a distance of 24 meters, this gives a horizontal velocity of 11.6 meters per second.  So much for an initial v of 9 meters per second!

This mid-flight bit tells us that they're using a quadratic model of drag, with a drag coefficient of .5 (that seems reasonable, given some cursory internet research and the drag coefficient of a smooth sphere, though I see someone suggesting that it might be as low as .25), and a lift coefficient of .15.  They're modeling the lift as being due to the Magnus force, and using the second expression from the Wikipedia description on the graphic.  I saw the first in school, and I'm not clear as to whether these are equivalent or not - anyone help me out here?  In any event, this effect's likely to be small, so we'll omit it from our analysis.  I can model the drag part with Interactive Physics, but it's not going to be too much of an effect, I think.
 OK, in this one, the ball's very nearly there - only 20 cm to go, reportedly.  They give us a velocity of 12.6 meters per second (again, if the initial velocity was 9 m/s, this isn't possible), and I get an angle of 34.3 degrees using Tracker (I tried to keep the lower line parallel to the long axis of the court - perhaps not the correct way to reference?).  The angled perspective would tend to increase my measured angle, I think - correct me if I'm wrong.

I got an angle of about 48 degrees at the beginning, so there's certainly some perspective issue.  The launch and catch heights aren't all that different, so there's considerable uncertainty in the angle measurement.  Taking this 12.6 meters per second and the two angles, I get 10.4 meters per second and 8.43 meters per second as the horizontal velocities.  The lower angle is closer to what I get using the original video's time, but I'm still quite suspect about the horizontal distance claimed. 

Using the flight time of 2.069 s (from the original video) and assuming a .75 meter change in elevation between the beginning and the end, I get an initial vertical velocity of 10.5 m/s (10.1 m/s if you assume that there's no vertical displacement). 

Combining this vertical velocity with the initial horizontal velocity from the total distance and time, I get an initial velocity of 15.6 m/s, at an angle of 42.2 degrees above the horizontal.  The angle's reasonable, given my measurements and a visual inspection, but I'm not sure where the velocities given in the video come from - neither seemed to make sense.

Let's look at the motion map-like tic marks on the path:
 Analyzing this first part in Logger Pro, I get a quadratic fit, though I wasn't able to get good agreement with the quadratic using the values of initial v, angle, etc. 

I also extracted the x and y positions (this came from a still image, since the speed is played with during the video, so I don't know the times, though I could count all of the tic marks and calculate - I'll leave that to someone else :) ) and created a dummy time variable to graph them against (just the point number).  You could check the scale again by matching up the curvature or the x-velocity, etc., but  I was satisfied to see a linear x graph and a quadratic y graph:
It's a cool commercial, but there are some bits of given data that don't seem to match up with the original video and the dimensions of the court.  Anyone have more analytical light to shed? Rhett?

Let me know what I've missed or miscalculated - it was a bit of a sprint during the game tonight!

Wednesday, March 14, 2012


What can you do with this? There's a lot of information packed into this photo!

Monday, March 12, 2012

Capstone: Vowel Resonators

This isn't officially a capstone (I'm only doing them officially with my AP class), but it's a great project that she did to demonstrate her Advanced-level understanding of timbre and FFT analysis.

Inspired by a website, Grace constructed some of these vowel resonators and compared the aural and FFT results to those in the article and to recorded vowel sounds.  After that, she revised her resonators a bit, using reeds made from straws instead of the duck calls, and decided that these sounds closer to the original vowels (the duck calls were too bright).

Read the paper here!

It's also posted at Capstone Learning.

Wednesday, March 7, 2012

Modeling Shadows - Application, Extension and Success!

 Last time, we began our process of modeling shadows - how and why they're formed, why some parts of the shadow are totally dark and others aren't, and began using diagrams to try to predict shadow composition and size.  The initial diagrams had way too many rays and were difficult to interpret, so they had reading on what we teachers would recognize as "the normal way" to ray trace.

In the past, we'd go through that first day of experimenting and fumbling, creating less than perfect diagrams, etc. all together, with me doing the drawing, and them doing the note-taking.  I've even had them do the experiments to determine the trends, but then when the diagrams started, it was all me.

It was quicker, but they surely couldn't do all of this during the next 45 minutes of class:

Today, they:
  • Ray traced an actual-size model of a ball/bulb/wall system
  • Determined where the shadow was dark or lighter (remembering the names, too)
  • Drew, on a smaller board, an actual-size prediction of what the shadow would look like
  • Checked that prediction against the demo - success! 
  • Made an accurate prediction about how this would all be different with a point-sized bulb using ray tracing
  • Connected the shadow parts to the types of solar and lunar eclipses
  • Determined where the antumbra was, what the light looked like from there, and what that shadow looks like on the wall
Modeling FTW.

Even when I've had kids do the reading before class, they weren't able to actually apply ray tracing (if they in fact learned the mechanics by reading) and understand what it tells them.  That bit about drawing what the shadow will look like on the little boards?  That was a totally new perspective on shadows from the ray tracing they've done, but they were able to do it without skipping a beat.

It's the sequence, and who's doing the work that's the magic.  They modeled shadows last class. They  came up with a (moderately effective) ray tracing method, so they saw the need for a better one. They did some reading with a purpose and enough background to assimilate the reading (assuming that they did the reading - one section almost uniformly did, and everything was awesome.  The other section didn't read at a high rate at all, and it wasn't nearly as rosy.  Nothing works if they don't!).  They made predictions and saw them validated. They did it.  The result? They know it.

Monday, March 5, 2012

Modeling Shadows

Today we turned our attention to shadows, noticing first that we could get shadows of different sizes and compositions (some are really dark, some are really light, some have a dark nougaty core surrounded by a lighter halo...). 

How could we predict what kind of shadow we'll get?

We listed variables that might help determine the shadow size and/or composition:
  • Bulb/ball distance (our objects were balls)
  • Ball/wall distance (the shadows were cast on walls)
  • Bulb size
  • Ball size
  • Bulb brightness
  • Ball shape
  • Bulb shape
  • Alignment of ball/bulb/wall
This is by far the biggest list of variables that we've had so far, so they started to narrow down a few:
  • They decided to stick with the "normal" incandescent bulbs that we had for the moment
  • We'll stick with the 2.5" styrofoam balls on little wire stands for now
  • They decided that the bulb brightness wouldn't really do much, and that we didn't really have a way to vary it anyway 
  • We'll stick with a linear arrangement - ball, bulb, wall in a line, level with each other
That basically left us with the ball/bulb and ball/wall distances - we needed to determine their effects on the shadow size and composition.  How do we do that?  How many experiments will we need?  They landed on four experiments, controlling one distance and varying the other, and measuring/describing either the size or composition.  It's a nice number, since we have four groups.  Each group took data and presented their results to the rest of the groups:

Varying Ball/Bulb distance, examining shadow size
Varying Ball/Wall distance, examining shadow size 

Varying Ball/Wall distance, examining shadow composition 
Varying Ball/Bulb distance, examining shadow composition 

There are some great observations here, chief among them that the relationships aren't linear. With so many variables and nonlinear relationships, we went to a digrammatic method to try to make the predictions.  I didn't picture any of these, but each group was able to realize:
  • Light travels in straight lines
  • Shadow happens because (at least some of) the light from the bulb is blocked from reaching some location on the wall
  • Not all of the light has to be blocked to make a shadow, but if it is, it'll be that dark part of the shadow
  • Finding the dark part of the shadow (which we then named the umbra) wasn't too terribly difficult
  • Finding where the edge of the shadow was (we called the light outer part the penumbra) was much more difficult - at this point, we had a billion rays going every which way!
  • We needed a better method - one that required drawing fewer rays, for sure.  That's our motivation for doing our reading tonight.