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. F_{12} + F_{10} + F_{2}] so the miles would be approximately 89+34+1=124 [i.e. F_{11} + F_{9} + F_{1}].

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)t^{2} 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!)

Tags: Bones, exponential decay

## Leave a Reply