Time for some maths! Who likes maths? That, or “Who likes math?” anyway was a question that Jonathan Coulton asked the crowd when R and I saw him in Bristol a couple of years back. It was, of course, a prelude to his song ‘‘Mandelbrot Set” but it was a surprise – to me, and I think to JC – to find that I was almost the only person in the crowd to call out an enthusiastic ”yeah” in response. But I like maths a lot, which occasionally leads to trains of thought like this…
Game Spoilers ahead: when the players beat Skullmageddon in Double Dragon Neon, they knock him off a ledge where he falls for a long time (3 minutes, 32 seconds), singing the end credits song. The question occurred to me: how far does Skullmageddon fall?
Well, I’m going to cheat a bit: the alien-esque dimension that Skullmageddon calls home allegedly doesn’t support human life – so Billy & Jimmy magically transform into robots when teleported there, and the sky is a near-perfect star-field without clouds, so I’m going to assume no atmosphere at all. The lack of air resistance means I don’t have to work out the terminal velocity of bone/mythril composites, and can use simple kinematics to calculate the distance fallen. Also, Billy & Jimmy experience Earth-normal gravity in this environment, so I’m going to assume that the acceleration due to gravity here is equal to g.
Assuming a spherical lich in vacuum, the equation to calculate distance travelled (fallen) under a constant acceleration g is thus:
Since x0 (starting height) and v0 (starting speed) are both 0 it’s a simple bit of arithmetic to see that in 212 seconds (that’s 3 minutes 32 seconds), Skullmageddon appears to fall a staggering 220,450m and reach a velocity of gt=2080m/s.
Clearly this isn’t logical – starting at a distance of 220km above the planet’s surface, the battlefield at the top would experience significantly reduced gravity – and be closer to geostationary orbit than to sea-level. This is the point where I have to either start considering air resistance and terminal velocity, or admit it’s silly to bring physics to a discussion of video games.
Assuming that Skullmageddon’s dimension has similar atmosphere to Earth, the drag experienced by Skullmageddon due to air resistance is:
Where ρ is the density of the air, A is his cross-sectional area, C is his drag coefficient and |v| is the magnitude of his velocity. This drag force acts in the opposite direction to motion. He reaches his terminal velocity when the drag pushing him back up balances with gravity pulling him down. We can solve this numerically by stuffing “sensible” values into the equation.
I’m using estimates for surface area and drag coefficient stolen from this informative Wolfram article about the record breaking skydiver Felix Baumgartner: C=1.3, A=0.4m2. For the density of air, I’m using ρ=1.2754kg/m3, which assumes IUPAC standard temperature and pressure. I’m not sure that this is a valid assumption, but we’ll see where it leads us. I’m going to assume that the mythril from which Skullmageddon’s enchanted bones are made is roughly as dense as steel: 8000kg/m3. This is 8 times the density of water, and the human body is roughly as dense as seawater, so I’m going to assume that Skullmageddon’s mass is 8 times that of similarly-sized human. For a human 2.5 metres tall (Skullmageddon is very tall, and towers over his minions and the player characters), with a BMI of 20 (at the low end of the healthy spectrum) that human would weight 125 kg, so Skullmageddon’s mass should be a nice, tidy m=1000kg.
Stuffing all these numbers into our equation to solve for v gives us a terminal velocity for Skullmageddon of 172 m/s. That sounds fine to me – he’s slightly larger and much denser than an average human, so it sounds logical that his terminal velocity should be a bit faster. This means he reaches terminal velocity in about 17.5 seconds (whilst falling 1500 metres), and falls for the remaining 194.5 seconds at terminal velocity, falling another 33500 metres. In total, Skullmageddon falls for 3 minutes 32 (which we knew from the start), and falls through a height of 35 kilometres! Whilst that’s a long way to fall, it wouldn’t make a world record: Felix Baumgartner jumped from 39045 metres.
He didn’t get a world record for distance fallen, he didn’t beat the Lee brothers, and he didn’t get a date with Marian. Basically, it wasn’t Skullmageddon’s day. He did sing a frustratingly catchy and fourth-wall-breaking song on the way down, though, which I’m fairly sure can’t be said of Mr. Baumgartner.