Applying game theory to the game Rock, Paper, Scissors, and to other, less important aspects of life.
Chances are you’ve played Rock, Paper, Scissors, but how do you calculate your strategy, if you have one at all?
In Rock, Paper, Scissors: Game Theory in Everyday Life, physicist Len Fisher points out that putting yourself in your opponent’s mindset is a key to success in the game.
It’s all part of game theory, which has to do with everyday strategies and commonplace interactions — and not just those designed for winning at Monopoly or trapping wild elk, as it may sound. Fisher, a visiting research fellow in physics at the University of Bristol and author of several science books for lay audiences, argues that a teaspoon of this sort of thinking can illuminate a range of human behaviors. Not to mention that game theory offers a handy explanation of why all those teaspoons keep disappearing from the communal lunchroom at work. (Individuals think it won’t hurt the collective if they take “just one” spoon, but, voilà, in no time, there are very few, if any, left for the collective to use.)
While I was on vacation I had seen a couple of videos/links about a guy who launched himself with some water rocket of the large soda-bottle variety, and thought that this was the sort of thing Rhett would analyze over at Dot Physics, and as I catch up with my blog reading, I see that it is so: Water Bottle Rocket Guy
“Water bottle rocket guy” is too impersonal and too long to type repeatedly, so I will refer to him as “Mr. Payload”
The thing that screams, “Fake!” the loudest is the video snippet that indicates a cable attached to Mr. Payload’s harness. I notice that he also starts tumbling, as one might expect from a torque from the rockets, but this motion does not continue — something that a guide cable would interrupt. There’s also the trajectory analysis, which doesn’t jibe with expectations.
Rhett does a quick energy analysis of the maximum height, but the analysis assumes all of the energy goes into Mr. Payload and his rocket shell, thus giving an absolute maximum height, and the number he gets isn’t realistic. One must also consider the large amount of energy contained in the expelled water that generates the thrust to get a more realistic limit, as well as the energy used for the forward motion.
Rhett uses 1L of water per bottle , but to me it looks like there is more. I’m going to assume 20L of water but the same energy (i.e. higher pressure) and that Mr. Payload has a mass of 60 kg, which is more than Rhett uses but makes the math easy. Since the water is expelled quickly — it appears to be gone before he’s more than 2m above the dock, so I’m just going to model this as an explosion, with the water getting an impulse and Mr. Payload getting an equal and opposite impulse. Their kinetic energies must add to the total energy of the system, of 27kJ.
We have the sum of the KEs totaling 27 kJ, with \(KE = p^2/2m \) and equal magnitudes of momentum.
Solve for momentum, and I get 900 kg-m/s, or a speed of 15 m/s for Mr. Payload. If launched at ~30º, as in the video, that’s a height of under 3 meters, ignoring the considerable drag. It also means that about 20 kJ of the available energy (i.e. 3/4 of it) went into the expelled water.
One of the comments links to a video which looks real. The launch is at about 3:15
