## Sunday, March 18, 2012

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!