Every time I go sledding, I think about the gradient vector. You see, when you get your sled and go to the top of a hill, you usually pick a way that you want to go down -- avoiding the trees, for instance, and ending up at the bottom of the field rather than veering off to the left and going into the road. But you know what? You can't always do it that way, because it's all about the gradient.
Let me explain. If you have a surface -- a 2D object in 3D space, like a warped piece of paper -- the gradient vector tells you, at any point, what is the steepest way down. You can also use it to determine the perpendicular vector to any point on the surface, but that does not interest us here. So at the very top of a mountain, the gradient vector is 0, because it's flat and there is no steepness so there is no steepest direction. On a ski slope, the gradient vector points in the same direction as your skis.
And that, as you may have gathered already, is what interests us here. Because once you choose the point on the hill where you will start your sled run, the rest of the course is determined for you. Of course you can stick your heels into the snow to try to have some effect, but really, you cannot significantly alter the course of your destiny by such meek methods.
The sled will choose the steepest way down the hill because that is the most efficient. The higher you are on the hill, the more gravitational potential energy you have, and the more energy you have, the less happy the world is. The world likes everything to be in the lowest energy state possible; i.e., it wants the sled to be at the bottom of the hill, and the best way to get it to the bottom of the hill is to take the most direct, also known as the steepest, route.
So all you have to do is give me an equation of the form z=f(x)+f(y) for your hill (assume there are no overhangs, so that it can be expressed in this form), and a starting point (x,y) (or (x,y,z) if you're being nice), and I'll tell you the precise trajectory of your sledding expedition. Oh, and I'll need a starting velocity (probably not 0 in most cases, if you give yourself a push) and a coefficient of kinetic friction between the sled and the snow. But once I have that, I can tell you! Sounds easy, doesn't it? No? Well, gravity is just that smart.
Oh, and I went sledding today. It was really warm out, so I went out in just a T-shirt and running shorts, with snowpants, boots, and gloves. I offered a ride on the tube to everyone that went by, but alas, no one took me up on my offer. Pity. It was a beautiful day for sledding, and the snow will all be gone tomorrow. Here's hoping we get a really big blizzard for dead week.
5 hours ago