This is a story about projection...

Do you want to predict the future? Would you like to be in the right place at the right time? Ever wonder how ice hockey star Wayne Gretzky gets to where the puck is going to be, not where it's been? How does a torpedo fired from a submarine intersect with its target, while following a relatively straight trajectory? Welcome to the world of dead reckoning.

Imagine I have a single data point - a latitude and a longitude - which tells me the location of a target at one point in time. I don't know how fast it's going, and I don't know what direction it's going. Fairly useless information. In the parlance of physics, I know location, but I don't know momentum. All we can say is that the probability of its next location spreads out from the point of origin: the thing being tracked is very unlikely to be a long way away, and it's more likely to be somewhere near where it was last seen. This is a probability cloud.

Next, we've measured two data points. We're sure it's the same thing that we're tracking, because we have identified some unique characteristic about it. In the submarine example, the sonar operator will be listening to the sound of the propeller, so they will know if it's the same ship, because it's got the same number of propellors generating the same kind of sound, underwater.

If we have two data points, we can draw a straight line between them, and assuming the same course and speed, we can predict where the object we are tracking is going to be next. We measure the distance between the two points, and divide by the time it took to travel between them, and we have the speed. Then, we simply extrapolate the line between the two points, and we can guess where the target will be on that line, at a future time.

But, the simple extrapolation model is flawed. Perhaps currents are pulling the ship in a different direction from the direction steered. Perhaps the captain of the ship is steering an arc-shaped course, to go around an obstacle, or to change direction.

With three data points, we can now fit an elliptic curve to fit those measurements. The elliptic curve shows us whether the captain is steering or drifting right or left. Additionally, we know whether the ship is speeding up or slowing down. If the ship is accelerating, but steering a constant course, it will be further along the line we plot than we think it will be. Equally, if the ship is decelerating, it won't get as far along its course as we anticipate.

Finally, with four data points, we can come up with a mathematical function that fits the data. To give a very simplified example, if the captain of the ship is zig-zagging to avoid submarines, we could fit a sine wave to the data, and predict when the ship is going to make its next turn... assuming that the captain tacks with regular frequency, of course.

As we gather more and more data points, plotting the course history of the ship we're tracking, we can come up with ever more sophisticated mathematical models to predict where we're going to see the ship next.

The arc of a ball travelling through the air is fairly easy to model. We just need to model momentum, gravity and air resistance. Momentum would be constant in a vacuum, so if you whack a tennis ball with a racquet while floating in the outer reaches of the galaxy - where there is little gravity - then the tennis ball will continue on a straight trajectory, at the same speed, more or less forever. Back here on Earth, the tennis ball will be pulled down to the ground by gravity, and it will also slow down, due to air resistance.

Given that air density and gravity are more or less constant at ground level, where you expect to find tennis courts, we probably only need 3 data points, and we should be able to make an accurate guess as to where the tennis ball's going to land. Even with gusts of wind, the flight of the tennis ball is relatively short, so the wind speed is unlikely to change dramatically during the flight of the ball. It should be a case of simply fitting an elliptic curve to the 3-dimensional co-ordinates of the 3 points of measurement, and then working out where that curve intersects with the plane of the ground, to predict where the ball's going to land.

"But what about spin?" I hear you ask. Yes, good point.

Spin is angular momentum. For our tennis ball example: the fluffy exterior of the ball will exert a 'grip' - friction - on the air. Therefore, the spin of the ball makes a reasonable difference, as more air gets pushed towards one side of the ball, by the rotating face. Just like a wing, the ball will experience a small amount of lift: the ball is 'pulled' towards the side that is spinning in the same direction as the air flow. That's the Magnus effect.

If we had hit our tennis ball with spin out in the vacuum of space, it would not create a curved trajectory, because the fluffy tennis ball has no air to grip on - a vacuum is frictionless. However, the centripetal force of the spinning ball will cause it to stretch to be slightly bigger than it would be if it wasn't spinning, because the outside is being thrown away from the centre, like a centrifuge. The elasticity of the ball will be trying to pull it back inwards, and in so doing, energy will be released in the form of heat. Eventually, all the energy of the spin will be radiated away, and the ball will slowly stop spinning. This effect is far too minuscule to even bother calculating, for our tennis court example here on Earth.

Of course, all this maths is just to model a dumb tennis ball, which can't make decisions and alter its own course, once it has left the racquet. To predict the behaviour of even a single individual is hard - this is the work of top poker players - and when we try to model complex systems made up of vast numbers of individuals - such as financial markets - the systems become chaotic and unpredictable. Thus, past performance is not a predictor of future results, unless you're a tennis ball.

Tags: #physics