## Math in Bones

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So I’ve been watching DVDs of Bones, as you might have gathered from an earlier post. And one of the episodes from Season 2, “Spaceman in the Crater”, has three separate spots with some math! Not a lot of math, but it doesn’t take much to make me happy.

The first bit comes early on, when Special Agent Booth and Forensic Anthropologist Dr. “Bones” Brennan examine a body that’s fallen from the sky and formed a crater.

Brennan (immediately after commenting on the man’s loafers): He hit the ground at approximately 200 kilometers per hour.
Booth: How can you tell that by his shoes?
Brennan: 124 miles per hour is terminal velocity for a falling human.

See that fancy unit conversion? It’s right, too. (OK, not hard math, but for all we know Brennan used Fibonacci numbers to figure it out, since 200=144+55+1 [i.e. F12 + F10 + F2] so the miles would be approximately 89+34+1=124 [i.e. F11 + F9 + F1].

Then a little later, Dr. Zack Addy took it a bit further:

A human being reaches terminal velocity after falling 200 to 220 meters, depending upon air resistance. Velocity would be achieved between 5 and 8 seconds, depending upon atmospheric conditions, body position, and clothing. He fell from a minimum of 1200 feet. I can run through the math if you like.

Sadly, no one wanted Addy to run through the math so I can’t see what played into this. Presumably if air resistance were negligible (which it isn’t) and if the body was dropped rather than pushed, then it would fall ½(9.8)t2 meters after t seconds; that’s 122.5 meters per second after 5 seconds and 313.5 meters per second after 8 seconds. The upper bound of 8 seconds makes sense to me, since air resistance would slow the speed down, but I’m a little surprised by the lower bound of 5.  Shouldn’t it be at least 6 seconds, and probably closer to 7?  Maybe Addy is thinking the body might have been pushed.

In the final math segment there’s a bit of exponential decay. Addy shares the following observation a bit later, after they’ve determined that the victim was an astronaut:

Astronauts lose 2% of their bone mass for each month spent in space. Our victim’s legs, hips, and lower vertebrae have demineralized over 20%, indicating 10 months in space.

Using that 2% per year, the amount of bone mass left should be (0.98)n after n months. Solving (0.98)n=0.8, in order to find out when the boned demineralized 20%, leads to n=log(.8)/log(.98) [where the log is to your favorite base], or n≈11.04 months. That doesn’t fit with the 10 months mentioned above. In this case, though, rounding could be the culprit: if it was really 2.4% of bone decay per month and that number was rounded down to 2% for convenience, the formula becomes (0.976)n=0.8, giving n=log(.8)/log(.976)≈9.2, which fits with the data. I’ll grant Addy this one.

(And yes, the rest of the episode was good too!)