That makes no sense. Lets call the hill the x axis. Further to the right, the positive direction, on the x axis is downhill. If we start the skiers at 0 on the x axis, the thought experiment is valid. Why would you start the skier in the air at some point other than 0? There is no reason to do so; you’re just giving one skier a comparative advantage that negates the point of the exercise.

The other thing is that the hill does not magnify any forces. The skier in the air is being accelerated straight up and down. Newton tells us that an object not being accelerated by a force remains at the same speed. When you are standing on the ground, the force exerted by gravity is exactly matched by upward force exerted by the ground. You do not accelerate.

When you jump, the force from the ground vanishes the instant you stop exerting force against the ground. You are now accelerating 9.8m/s/s down. This force is unbalanced, thus you accelerate - your initial velocity gets scrubbed off, you stop, your velocity reverses until you hit the ground again, at which point the forces are in balance again, and you stop moving.

Now, if you are standing on skis on a 45 degree slope, gravity still pulls straight down. But the ground pushes up at 45 degrees (perpendicular to the hill) These forces are *not* balanced, thus you start accelerating in the x direction – your velocity steadily increases (until you reach terminal velocity).

This “relative to the slope” thing is a red herring. The only “ahead” is further on the X axis. By placing one skier at x=0 y=10 and one at x=1 y=11, you’re simply putting a skier ahead arbitrarily. The only way to win the race is to get to the X position of the finish line (call it x=100) first. It doesn’t matter if you cross it at y=100 or y=0.

Let’s say both skiers are going 10 m/s in the X plane. Skier A is in the air, Skier B is on the ground. So far they have the exact same time in the race, and A has just jumped while B managed to stay on the ground. Let’s further say that the slope of the hill is such that the horizontal acceleration is 5 m/s/s (roughly 45 degrees). Ignoring air resistance for now:

Skier A 1 second later has an X velocity of 10 m/s. 2 seconds later he has an X velocity of 10 m/s. 3 seconds later he lands, and his X velocity is 10 m/s. At 4 seconds, he is moving 15m/s. He does not accelerate in the X direction during his jump, because the only force acting on him is in the Y axis (which is pulling him down at 9.8 m/s/s)

His total distance over those 4 seconds is 45m.

Skier B is moving at 10/s in the X direction as well. At second 1, he is moving at 15m/s. 2 seconds, 20 m/s. 3 seconds, 25m/s, at 4 seconds, he is cranking out 30m/s. His total distance is 70m and he is going twice as fast in the X direction.

The key is simply that while skier A is in the air, his X velocity, *the only one that matters*, is not increasing. The Y velocity does not matter because the average Y velocity during the race does not have any effect on your finishing time.

So, a skier sticking to the hill which is 100 m long will travel down 50m, and a skier with an inspector gadget helicopter hat traveling straight forward will travel 0 in the Y direction. Skier A’s overall velocity will be higher if they tie, but his x velocity will be the exact same. This is where I think you are running into problems with this idea of “relative to the slope”. You can also imagine the helicopter guy moving up and down in a sine wave at 200 m/s as he travels forward – his overall velocity will be much, much higher, but the x component remains the same, and they still tie. If either skier has a higher X velocity, they win, no matter what their Y component is doing.

Thus, as was said above, sticking to the ground is faster than jumping, because you only accelerate forward while you are on the ground, not while you are in the air.

ETA: **Chessic**: the point is it doesn’t balance – you don’t get a boost forward from jumping except in video games. See the math in this post.