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I'm sure that, intuitively, the idea of a trajectory is clear enough. Maybe the first example you think of for such a beast is the path of a rocket to the moon, or say, more close to home, the motion of a ball hit or thrown by your neighbor. The path of a tennis ball, for instance. Or a football or baseball. You can also have the path of an aircraft or a racing car or a ship at sea. There are many other trajectories, as well, like the path you yourself might take when you cross the street or go to the dentist, or the way you run to intercept a tennis ball to hit a return shot. Even the price of a stock on the securities exchange is a kind of trajectory -- it doesn't move in space, but it reflects the way the price "moves" over time, and you can even "run" to intercept it at a certain price, assuming you predict its price action correctly. Anyway, that's what I see trajectories to be. 

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Okay -- so what? What's so special about trajectories. We know they're about as common as the air we breathe, even if you don't count the price histories of stocks and bonds.

Yes, well, the point I'm trying to make is that they can be central to learning skills.

In tennis, for instance, knowing about ball trajectories can make reading and hitting them a lot easier; if you know the weird moves they make, you can even anticipate what they will do.   Likewise, being able to judge the direction and speed of cars makes it a lot easier to cross the street safely. All in all it's a track and intercept problem at the tennis court, or a track and don't intercept problem on the street. The point is, the more you know about trajectories the easier it is to deal with them and the more confident you can be in performing the skills -- like hitting the ball or avoiding the cars.

In tennis you have to track and intercept the ball to put yourself in a good position to return a clean and hopefully a winning shot. To make a good job of it, it helps to know how fast it's going, the upward angle it has to start, where you can expect it to land, how high and fast it will bounce, and other stuff like that. It's a bit like selling widgets -- to be good at it you really have to know your widgets. You must know what the bloody ball is going to do. You have to understand how it moves.

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Let's face it -- the object in tennis is to hit winners -- either that or to set up winners by making your opponent do what he doesn't want to do, like hitting a weak return that you can smash. That's pretty much the way it is in sports, generally, and quite like what we do in most other activities in one way or the other, whether we like to think about it or not. However, you also have to know what to do when you effect the intercept.

To do this in tennis, for instance, it isn't enough to track and intercept the ball correctly and accurately. In addition, you have to know exactly how to hit the ball when you get to it. That is, you must know how your racket should meet the ball. You need to understand the ball-racket dynamics. You've got to know how fast to swing the racket, the direction the racket should have when it meets the ball, and the orientation angle it must have at the point of contact.

The thing is, the ball is coming toward you in a specific way and you want to return it in some special way. It's the dynamics that dictates what you must do. The racket is the only contact you have with the ball during a rally and the only control you have over where it will go. So you've got to understand its funny motions.

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