I'm as good at photography as I am at skiing, so since I sometimes take a stab at the skiing thing, here is an answer. As stated above, digital SLR's would be great for such pictures although things are changing fast and the entry point is not 1500 for a body, it is more like 800. A decent lens might be another 3-400 for one with plenty of zoom and not necessarily huge in size. This would get you in the game, not crystal clear to a pro's standards, but indiscernable to an amateur. Of course if you ever go to a photography forum, you will see that there are gear snobs there who would laugh at this setup, similar to the treatment my pocket rockets often receive here.
The picture is more about the capture than the camera. Sound familiar? It is possible to get such a picture with lesser cameras but there are many challenges. First is lighting. You have to find a spot with good lighting to allow capturing the blue sky and the face of the skier without one or the other being completely lost.
The second challenge is framing. Most pros would take the picture in a burst of multiple pictures. An amateur would try to time a single shot. It is possible to time a single shot, but you will miss at least as much as you hit that way. The biggest challenge is the delay between hitting the button and taking the picture. As mentioned, autofocus can be a big part of this problem so manually focusing first helps. Cheaper cameras still have significant delay even when manually focused. It is predictable, so you can time the shot with practice.
Some cameras cheaper than the SLR's do take reasonable bursts of pictures, sometimes more than 3 frames per second. The problem is that if you take the burst, the individual shots depend on the timing of the first and the interval in between. With fast action, it is possible to miss that way as well. That is why pro equipment rattles off rapid bursts of pictures. The other problem with a non-SLR is that once you hit the button, you are flying blind since the electronic viewfinder usually blanks so it is not possible to track the skier by looking through the camera. It is possible to run a burst of pictures and track the skier by panning on a tripod, keeping zoomed out to improve your odds, and cropping the image on your computer to zoom in. In other words, you can get the picture with a cheaper digital and have satisfactory results, but it will take some practice to get the picture with any sort of consistency. Of course, even the pros probably shoot a hundred pictures for every good one like that.
So if your goal is to try to create some photographic aids for teaching, it can be done but plan on doing multiple runs and shooting lots of pictures. The better equipment can improve your odds of getting good pictures, but it still depends on your ability more than the camera. The better equipment will also improve the quality of those pictures, if you were planning on supplementing your income with a side business in ski photography.