Splat!

Figuring out where a spatter came from is useful sometimes. Not in any field I've personally worked in, but then I don't usually work with things that go splat. Some things which go splat, where the spatter marks remaining after the fact are the only evidence available to figure out how exactly it happened, include volcanoes (which can make very big, very dangerous splats most sane people wouldn't want to watch in person) and people being attacked (which often ends with the source of the spatter in no condition to describe the attack).

One obvious thing about spatters is that the individual marks are ovals, and they point in the direction of their source. This has been known for a long time now, and has been used in forensics to determine where a victim was. It could also be used for volcanoes, if nobody saw which of the vents erupted due to running for their lives.

What the oval spatters didn't accurately point to was how high the source was.

This can make a difference in an attack situation, or perhaps more in the trial afterwards, when the forensic investigation results are used as evidence. A simple example: was the victim sitting or standing when they spattered? This isn't just academic; it could make the difference between calling it attack or calling it self-defense.

They could get an angle of impact using the ratio of length to width of the spatter marks, but tracing those lines back to their convergence point doesn't account for the arc objects make when flying through the air, and thus consistently overestimates the source height. Acceleration due to gravity and all that:

Some physicists decided to study this issue to find out if there was a way to determine the height of the source based on the spatter pattern. Obviously, they had to make spatters from a known height and then study the spatters to see if they could make the height come out of some correlation. Just as obviously, they didn't want to create the spatters in the usual way…

So, first they built a spatter-generating machine they could use to make spatters from a controlled height and launch angle and found a fluid that wasn't blood that would spatter in the same way. I'm not sure how many different things they tried before finding the winner, a mixture of Ashanti chicken wing sauce and Ivory dish soap.

After applying a bit of trigonometry and the constant-acceleration trajectory equation, they found that mathematically:

\[\tan\theta_I = z_0{2\over r_I} + \tan\theta_0\]

Since \(\theta_I\) is the impact angle on the floor (measured) and \(r_I\) is the distance from the source (measured) that leaves two unknowns, \(\theta_0\) and \(z_0\), the launch angle and source height, respectively.

That equation also follows the format y=mx+b, the equation of a line.

After making a lot of messes and measuring \(\theta_I\) and \(r_I\) for the resulting spatter marks, they plotted \(\tan\theta_I\) (y) against \(2\over r_I\) (x) for each measured drop mark and got a bunch of points that were arranged in a line, just as the math had predicted. The slope of that line, m, was \(z_0\), and the intercept, b, was \(\tan\theta_0\), and was pretty close to what they'd actually done with their spatter-generating machine.

I love it when math matches experiment so closely. The best part is, this method is inherently error-detecting and thus protects itself against wrong answers. If there is too much variation in source height and/or launch angle (multiple sources mixed together, for example), you'll get a random blob of points on the graph instead of a line, and thus won't be able to get an incorrect answer out of it.