Physics of Water polo

Abstract:

Water-Copters, Beckham and dirty playing 
oh right, and drowning just a little bit 

It has been some time now since I've written my last real post, even though the "passing remark" about butterfly flatulence was rather lengthy...

When I was fifteen or so I played Water polo at a semi-pro league, which is to say the only non-pro league in Israel. Now, for those of you who are not familiar with that sport, you don't have to be ashamed, this is NOT as embarrassing as for instance not knowing who Michael Jordan is, since basically water polo is as obscure as for instance Tolkien's "Mr. bliss". that is to say, in essence only people who play water polo or have an immediate relative that does, actually know what water polo is.

If you ask the average Schmoe what water polo was, you'd get something like "err... a bunch of people in water passing a ball around?", or otherwise simply - "you mean water basketball, right?"

Consequently, it's not surprising that if you search the web for "physics of water polo" you'd get very few resources and pretty lame at that. for instance the "eggbeater kick" entry in Wikipedia explains a little what that move is, but none of the physics involved is explained.

So, in a lame effort to contribute something unique to the web, and maybe get a higher rank in google's algorithms, this post will deal with water polo and physics.

The Water-Copter
Water polo players are probably among the best cardiovascular athletes in the world, that due to the fact that this game has 4 quarters, each quarter is theoretically 8 minutes long but about 12-15 minutes of real time, meaning the players are in the water for almost an hour.

That in itself is meaningless but you have to take into account the  "water-copter" move or the eggbeater kick, which is - moving your legs in two separate counter circles, which creates a constant water current directed downwards.

The "water-copter" technique

this move acts on the same premise as your basic helicopter, hence - the water-copter.

So how a helicopter works? simple! (well not that simple but whatever...).
each second the helicopter is supposed to accelerate downwards in a constant acceleration \(g\), that is, the gravitational acceleration. so starting from a stationary state in one second it is supposed to develop a downward momentum of \(m_{\chi}g\), where \(m_{\chi}\) denotes the helicopter's mass. so, in order to keep the chopper flying, the rotor moves the air downwards in a momentum that is equal to the downward momentum, thus momentum-wise we even out. more explicitly:
\[\left\{\begin{array}{l}m_{\chi}\cdot g \cdot 1sec =m_{air}\cdot V_{air}\\
m_{air}=\underset{Surface}{S}\cdot\underset{height}{V_{air}\cdot 1sec} \cdot
\underset{density}{\rho}\\
\Rightarrow V^{2}_{air}= \frac{m_{\chi}\cdot g}{S\cdot\rho}\end{array}\right.\]

So for a chopper that weighs about 7 tons (for instance AH-64 Apache) this means the air velocity going down should be about \(18 \frac{m}{sec} \approx 67 \frac{km}{hr} \approx  41 mph\) which is quite amazing.

and if you've ever seen a Black-hawk approaching a ship's aft for a landing or extracting approach, the sheer awesomeness of beholding this 10-ton helicopter "raising the sea" is simply beautiful, with a downward wind-blast equivalent to ~ 45 mph.


Anyway, I digress...
with the same kind of dynamics happening underwater, the water-polo player's mass is effectively decreased by buoyancy and thus we get that the water velocity under the player amounts to about  \(6.5 \frac{m}{sec} \approx 23\frac{km}{hr} \) which is formidable indeed. especially when you consider top running speeds.
this means that these guys get to underwater velocities that are equivalent to top 5k competitive runners.
taking into account water resistance being lower then track resistance we get to the inevitable conclusion that these guys develop speeds that are probably comparable to top track runners of 800 to 1000 meter runs.

the thing is, these guys have to do this just to stay afloat, not even mentioning lifting up to deliver a shot, so you can imagine how hard this sport is, when these guys are required to do the equivalent of running 4 5k runs back to back at an Olympic pace, to do nothing more then "stand" in the water, like sitting ducks if you will...

Bend it like Azevedo (or Sapic)
There's actually a movie I never saw that's called "bend it like Beckham", I don't know, maybe it was the stupid name, or maybe it's just the fact that I never really liked soccer, much to my father's disappointment.
he sometimes says "God! how is it possible that I raised two boys that don't like soccer?!?" and proceeds with "Are you guys sure your mine?" anyway, my sister actually used to like soccer very much, but me and my older brother - not to much, we enjoyed causing chaos and mayhem while trying to break each other... by the way over the years we've gotten pretty good at this, so now we don't fight as much...

Anyway, Azevedo and Sapic are both professional water-polo players, Azevedo was called at one point "the Michael Jordan of water polo" while Sapic was hailed as the best water polo player ever for some time.

But this part will deal more with the "bend it" and less with the "Beckham" -

If you ever played table-tennis, or plain ol' tennis or even basketball, you know that when you pass or shoot or whack away at the tennis ball, while applying SPIN, the ball behaves funny when bouncing off the ground...

A similar thing happens when applying spin to a ball simply going through the air!!!

Warning: Physics ahead

<physics>
Consider a ball going through the air in a velocity \(V\), and spinning away at angular velocity \(w\), like this here poorly executed diagram:

Well, there are a couple of things happening in this diagram, and I'll spare you the Navier-Stokes equations, cause they're a drag... literally :)

what happens is this:
essentially drag is proportional to the velocity, and so the drag on the lower side of the ball in this diagram is more pronounced than the drag on the upper side, ultimately, this means that the ball is turning downwards (in this diagram), or more generally in the direction of \(\overrightarrow{V}\times \overrightarrow{w}\).

the other thing that happens is the air in front of the ball is a tad denser than the air behind it and so the drag on the leading edge of the ball is more pronounced than the one on the late edge thus creating an additional effect in the \(\overrightarrow{V}\times \overrightarrow{w}\) direction.
This corresponds to the \(\eta\)  factor in the N-S equation, which is the factor that embodies the density of the medium, but the ball has to fly in really high velocities for this dynamic to be anywhere near being pronounced.

So that's what happens to a ball spin-flying mid-air, but what happens to a spinning ball when bouncing of a hard surface?

Well, for starters, if the ball bounces off a surface such that the spin is not perfectly perpendicular to the plane of incidence, that fraction of spin will simply give the ball a momentum component that is opposite in direction to the direction of the spin on the hitting edge.
For instance - if you spin the ball clockwise and it hits a wall the ball will bounce back and to the left, or in another instance like in this video, you apply a spin that goes in the direction of the ball on the upper side of the ball, and opposite on the lower side, thus when hitting the court the spin component donates additional speed to the ball after bouncing back from the court.
This move serves to trick the opponent, since we constantly gauge the approach of the ball and extrapolate where we should hit, but when the ball accelerates mid-shot, it throws off your intuition.

But, what if the spin component is perfectly perpendicular to the incident plane?
Well then, this case get's more interesting doesn't it? in this instance we have to turn to analytical mechanics and the deep understanding of a two-spring system which isn't that complex but I wont bore you with the details...

suffice to say that if we have a system of two springs, like so:

Two spring system

 We have two separate modes of oscillations, one is the combined up&down oscillation that would happen if a fat kid stands smack in the middle of this contraption and jumps up and down, and the other is the normal seesaw action we all know and love.

And so with that in mind, and the understanding that this is a nice model of what happens to the hitting edge of the elastic ball that bounces, we approach this dilemma.

So the ball approaches the incident plane in some kind of an angle, meaning that the leading edge hits the plane first and starts the double spring system, so we have the first mode contracting, and the second mode starts with the front "spring" contracting. since we have the same spring constant for both our springs (in this model) the cycle-time for both are the same, so the leading edge experiences the overall contraction (1st mode) PLUS the seesaw contraction (2nd mode).
the back edge experiences the seesaw contraction (2nd mode) when the overall mode (1st mode) is basically starting to extend thus the pressure on the back edge is significantly lower than that on the leading edge.

thus the normal force experienced by the edges are different accordingly, and thus friction isn't uniform leading to an overall addition of momentum in the direction of \(V\times w\).
<\physics>

So all in all the conclusion is that if you spin a ball in a right-hand spin (rightmost edge going forward) you get a swerve right, and if you spin left, you get a swerve left.

AND THAT"S HOW YOU BEND IT!!!!


Plain Ol' dirty playin'
Another great experience that is (for me) connected with water polo is the one of almost dying (again).
it turns out that water polo is also one of the most violent sports ever. it's probably comparable only to rugby.

We'll start with the amazing fact that, whatever the ref doesn't see, doesn't exist. bearing in mind the refractive nature of even still water, and taking into account these are less than still waters,  well, let's just say most fouls and injuries do NOT get noticed.

A typical water polo foul

and we'll finish it off with the unwritten law of water polo: if you dunked the ball, i.e. pushed the ball into the water, well, basically they dunk you. by the way, don't matter if it's your team or not, they'll dunk you!

Now, I didn't know that, and as a young kid, tired from all the egg-beating, I hung down on the ball for a second and dunked it. just a tad bit! I swear!!! when all of the sudden my own team member came up on me from the back and dunked me like there was no tomorrow, resulting in my almost dying, yet again....

Water polo at it's best :)

Fun fact: usually when people die from drowning they don't die from water filling their lungs, what usually happens is called "Laryngospasm" meaning the muscles in your throat contract violently, and you actually die from asphyxia.

This usually happens either from panic or due to swallowing hefty amounts of salt water...

So there you go, a "short" post with very few equations and a lot of physics, and best of all - another almost death for me! :)

Next time: Physics of floating stuff - submarines, boats, crowns and other stuff :)