I had hoped it would be obvious from context that in the falling scenario, we were only focusing on the period before hitting the slope.
The post was merely an attempt to get people to accept the result, which is a step I think is important before one launches into explaining, in physical terms, why it is the case.
EDIT: I think the explanation was confused by setting the exploration of rampyslopeyness on top of the ski course, I intended it to be an abstraction illustrating what sshould be intuitive: you only go faster towards the finishing line if you are touching a slope.
This can’t be true in general. The person with the highest average velocity from “Start” to “Finish” (i.e, along the slope) will have the best time. This is not the same as +X. The horizontal direction has no more relevance than any other to this problem.
Absolutely. This is the crux of the disagreement, I think. But when people say that a skier is travelling at 60mph, they mean 60mph along the plane of the slope. Not horizontally, vertically or any other arbitrary direction.
Once you accept that it is motion in a direction parallel to the plane of the slope that counts, the rest of the argument falls into place. It can then be seen that a skier dropping vertically is accelerating relative to the direction that matters, just like the skier on the ground.
Sigh, no, that isn’t right. A skier flying in a loop at 500 mph for ten minutes has an average velocity of 500 mph. But that skier hasn’t crossed the finish line. He will lose the race.
Alternatively, a helicopter in level flight will have no vertical component to it’s velocity, but will win or lose the race depending only upon it’s velocity in the X direction.
This is not a difficult problem, and it’s been explained quite well, with actual physics and math. I don’t care how fast a skier is going down the hill; that’s irrelevant, as has been explained.
It’s also completely obvious to the practitioners of any speed sport where getting air is possible. Rally racing, motocross, skiing, mountain biking, etc – you don’t accelerate in the air and even if you do convert some small amount of -Y force into +X force, it’s less than you would have gained were you simply accelerating the whole time by staying on the ground.
You don’t seem to understand vectors or the concept of average velocity very well. The skier’s average velocity is only 500mph if you compute it over an hypothetical straight path. If he flies in a loop, his average speed, from start to finish, is 0 mph, even though his instantaneous speed is 500mph along the loop.
(image showing a slope with two alternative interpretations of the finish line of a ski slope, one perpendicular to the slope, one vertical)
You appear to think that the objective of a ski race is to be the first to pass through the plane indicated by the red line in the diagram, i.e. that horizontal motion is what matters.
Those of us on the other side of the argument believe that the winner is the person who first passes through the plane indicated by the blue line, i.e. that motion parallel to the slope is what matters. A skier could have a higher average horizontal speed but still not reach the blue line first.
I am quite sure that by the book the finish line is the line on the ground, so in your diagram that’s the point where both lines intersect. Anyway that is not very germane to the argument and will only serve to confuse the issue further in my opinion.
I am uncomfortable with the idea that the objective is to reach the finish line, because skiers can be airborne when they cross the line. Strictly speaking, the objective is to pass through a plane which includes the finish line; otherwise would a skier who files over the line not count as having finished the race?
But yeah, I guess it’s a minor point, and it’s clearer if we think of it your way.
Boys, boys, boys, I want you to each go to your kitchen table and prop it up on an angle. Rest one egg at the top of the table, and hold another egg beside it over open air. Let go of both eggs at the same time. Carefully watch which egg travels the furthest laterally.
Let me be clearer. Obviously the choice of axis can’t have any influence on the underlying physics. My position is that everything balances out in terms of acceleration (ignoring friction), except that (usually) you are taking a longer path from A to B by taking air (unless there is for example a sharp dip in the slope). By not taking air your direction of travel stays closer to the shortest path from start to finish, without any significant drawbacks. What you gain in horizontal motion with the “push” of the slope you lose in vertical motion (compared to an airborne skier), which is just as crucial.
The other disadvantage is that the loss of steering will tend to put you off the ideal trajectory around a bend.
Also if you land hard, that is at an angle close to perpendicular, there is a large dissipation of energy in friction forces and dampening, but this is usually not an issue in professional skiing.
The aerodynamic drag of airborne travel vs friction on the slope is largely a wash I suspect.
I wanna be like you, with a kitchen floor at a 20 degree angle. Of course I’d get over it in a couple weeks but at least I’d know what psychedelic drugs felt like at that point.
I wonder if skiiers/boarders ever inadvertently give up speed as they come over ramps to avoid “air”, and what the trade-off is? Someone who hits the ramp going 35 mph and takes on a lot of air is still going to come out ahead of someone who hits that same ramp at 25 mph while staying close to the ground. But what about 32/28? 31/29?
Ignoring the skier control and aerodynamic factors and just focusing on the acceleration part of the air…
There are two main details to this problem that have been said but not properly emphasized, namely (1) that the speed difference depends entirely on losses and (2) that the total distance traveled differs along the two routes.
Consider this picture
\A
\
\
\__
\
\
\
\B
\
Put two skiers at point A and let them start sliding down. One will take the little jump in the middle and the other will not. They are both back on the slope before point B. Ignoring losses for a moment, energy conservation ensures that when the skiers cross point B (possibly at different times), they will have the same velocity. That is, all the kinetic energy they have accumulated has come from the loss in elevation from A to B, and that is the same for the two skiers.
The only differences you can have in the speeds at point B come from losses. The total acceleration, whether coming gradually or all at once at the landing, balances out in the lossless case. With losses now: for the non-air skier, the unique losses are ski-on-snow friction. For the air-bourne skier, the unique losses are the inefficiency of transferring perpendicular (to the slope) speed into parallel speed (with energy absorbed by the legs and sent into the ground.)
One of these is going to be dominant, and those who have skied a bit will likely agree that landing is the bigger loser of energy (especially for Olympic skiers with beautifully maintained ski surfaces.)
The other issue is total travel time. The airborne skier travels a greater distance and also never has more parallel velocity than the non-air skier and has for some period a smaller parallel velocity. Thus, travel time from A to B will be larger.
As for technique, you don’t try to loose speed to avoid air. Rather, you either hop over the lip or otherwise use your legs as shock absorbers to prevent the bump from pushing your mass upward. That is, you keep your torso on a smooth trajectory, and you just lift your legs just right so that your skis glide over the surface without taking any vertical force.
Aha! The source of your confusion. Motocross et al. is different, since the tires need to touch the ground to accelerate. This is the only way energy from the person or vehicle can be turned into kinetic energy.
Skiing is different. All the energy comes from gravity, which acts on you whether you’re airborne or not.
You also seem to keep forgetting to tilt gravity when you tilt your hill into the x-axis. The airborne skiier still accelerates sideways, in the x plane, because gravity is tiled down at a 135 degree angle (from vertical).
A
.
.
.
_|
B_______________________F
.
.
.
_|
As shown here, both skiers experience the same sideways acceleration due to gravity. The fact that B’s normal force cancels out his vertical is irrelevant.
So why’s it slower, vector-wise? Because of what happens pre-flight:
^
|
| _
__A_/ \_____________F
.
.
.
_|
Right now, A’s normal force is coming straight out of his head, resulting in a net forward force. But on the front of the bump, the normal force goes out of his head AND BACKWARDS, slowing him down for a moment. He makes up for this slightly in the lowered friction that he gets from going through air instead of over snow, but this is obviously not to his net advantage.
Add to that the fact that he now has to travel in a parabola instead of a flat line. THIS is the main reason, besides all that balance and air-crosssection stuff.