## Forked

Wow.  It’s kind of unsettling how quickly two weeks can pass in between posts.  I don’t know how Dr. Allain finds the time to be so consistent posting to his blog.

One of the television programs that gets watched on in my house is Minute to Win It.  You know, the 6 minute show they somehow stretch out to an hour with that annoying host with the terrible hair.  Fortunately, DVR allows the show to be watched in under 10 minutes, as God intended.

Anyway, a lot of the contests they have challenge one’s skill, balance, concentration, etc.  But when the stakes are high, the contestants are usually subjected to a game of luck.  A recent case in point was a game called “Forked”.  In this, the contestant is required to roll a quarter in between a fork tine from 16 feet away.

Now, this seemed kind of unfair, but I figured I’d calculate just how difficult it was.

First, we’ll see how hard it is just to hit anywhere on the fork.  Then we’ll check out how precise one needs to be to lodge in a single tine. This analysis isn’t so different from the field-goal analysis from an earlier post.  For our purposes, we’ll assume that once released, the quarter travels in a straight line.

According to the show, each tine is separated by $\tfrac{3}{16}''$.  I’ll assume that each of the tines is also $\tfrac{3}{16}''$ wide.  The total span is therefore $\tfrac{12}{16}''+\tfrac{9}{16}''=\tfrac{21}{16}''$.

What does $\tfrac{21}{16}''$ look like from 16 feet away?  We’ll there are a couple of ways to do this, but I’ll rock a trigonometric approach, using the tangent operator to calculate the angular span of the fork from 16 feet.

$angle = 2 tan^{-1}\frac{21/32}{192} = 0.39^{\circ} = 24\mbox{ arcmin}$

Where the units mean 24 arc-minutes or 24/60th of a degree.  I’m not sure what you make of this, but it seems small to me.  Compare that to the span of the moon and the sun in the sky; they each span approximately 30 arc-minutes in the sky.  So follow me here, but that means if we were to build a plank all the way to the moon/sun, it would be easier to roll a quarter to hit either of them (again, assume the quarter stays straight and can make it the whole way), than it is to hit the fork.  Holy cow.

What about actually rolling it in between the space between two tines?  Now, since the quarter has finite width (0.069″), the quarter actually only has $0.1875'' - 0.069'' = 0.1185''$ of play and still be entirely within each tine.

$angle = 2 tan^{-1}\left(\frac{0.1185/2}{192}\right)= 0.035^{\circ}=2\mbox{ arcmin}$

Ok, 10 times smaller!  Granted, there are three such openings in which you can lodge the quarter, but you’d have to aim less carefully to roll a quarter between Jupiter and Europa!