I add a rather dumb example too (please fill in the obvious blanks):
Suppose you threw tennis balls into the air, several balls each milisecond. Suppose you placed a jagged roof at hight h. Without the roof the tennis balls would travel to height 2*h. Suppose the jagged roof scatter tennis balls in all concievable directions. Would you accept that you get an accumulation of balls in the vicinity of height h compared to what you would get without the jagged wall?
What you’re doing by making the roof more jagged is relaxing what you mean by being ‘in the vicinity of height h.’ You don’t have a precise enough definition for that to be a well-formed question. The jaggedness means the roof’s height is not really a single number, it’s a range. We haven’t discussed either the specific roof shape or the distribution of the balls’ trajectories (and thus their horizontal momentum and their kinetic energy distributions). On colliding, a ball will either be deflected net-down or net-up, and in the latter case it will soon hit again, and again, until it deflects sufficiently net-downwards or until gravity reduces its vertical speed to zero. So, sure, when the roof’s jaggedness increasing its maximum height by some j<h, then on average the balls will stay in the air longer, and the additional time will mostly be spent between height h and height h+j. And because the vertical speed at height h+j will be lower (even for undeflected balls!) than at height h due to gravity, the fall will start out slower than you’d get from a perfectly elastic deflection from a flat roof at height h. If j is tiny, the roof can’t be that jagged, and so the effect on ball distribution will also be tiny. If j is large, with such a shape that many balls can actually make it significantly beyond h, then you can’t call it a ‘roof at height h’ anymore.
Suppose h=10′, and j=1′, and the jaggedness is set up in a way that makes the average roof height 10.5′. Then what you’re saying amounts to something like: Before the balls were vertically distributed quadratically, like if you’d had them following the usual gravitational parabolic trajectories but just truncated off all the time they’d have counterfactually spent in the top half of a height-2h room. But now the room is ~5% taller, and the balls spend nonzero time in the new top 5% of it, and we’re only truncating the top 47.5% of the parabolic trajectory on average, and we have added more ways for the room to interconvert vertical and horizontal moment.
Obviously I haven’t done any simulations or written down any equations to estimate the actual new distribution quantitatively. That would depend on the specific roof shape in ways I can’t easily capture in a simple equation (maybe someone else could, but I can’t). Even still, rephrased the way I put it above, that’s not nearly as surprising as it sounds when you stay vague and handwavy about it.
I suspect you could define a roof shape and a distribution of horizontal momentum vectors such that the balls would on average be deflected down faster than in the case of the flat roof.
Now, if I were to make the roof sticky instead of jagged, then sure, the balls spend more time there right at height h. But then the roof is absorbing the momentum and kinetic energy, producing heat in the process.
I add a rather dumb example too (please fill in the obvious blanks):
Suppose you threw tennis balls into the air, several balls each milisecond. Suppose you placed a jagged roof at hight h. Without the roof the tennis balls would travel to height 2*h. Suppose the jagged roof scatter tennis balls in all concievable directions. Would you accept that you get an accumulation of balls in the vicinity of height h compared to what you would get without the jagged wall?
What you’re doing by making the roof more jagged is relaxing what you mean by being ‘in the vicinity of height h.’ You don’t have a precise enough definition for that to be a well-formed question. The jaggedness means the roof’s height is not really a single number, it’s a range. We haven’t discussed either the specific roof shape or the distribution of the balls’ trajectories (and thus their horizontal momentum and their kinetic energy distributions). On colliding, a ball will either be deflected net-down or net-up, and in the latter case it will soon hit again, and again, until it deflects sufficiently net-downwards or until gravity reduces its vertical speed to zero. So, sure, when the roof’s jaggedness increasing its maximum height by some j<h, then on average the balls will stay in the air longer, and the additional time will mostly be spent between height h and height h+j. And because the vertical speed at height h+j will be lower (even for undeflected balls!) than at height h due to gravity, the fall will start out slower than you’d get from a perfectly elastic deflection from a flat roof at height h. If j is tiny, the roof can’t be that jagged, and so the effect on ball distribution will also be tiny. If j is large, with such a shape that many balls can actually make it significantly beyond h, then you can’t call it a ‘roof at height h’ anymore.
Suppose h=10′, and j=1′, and the jaggedness is set up in a way that makes the average roof height 10.5′. Then what you’re saying amounts to something like: Before the balls were vertically distributed quadratically, like if you’d had them following the usual gravitational parabolic trajectories but just truncated off all the time they’d have counterfactually spent in the top half of a height-2h room. But now the room is ~5% taller, and the balls spend nonzero time in the new top 5% of it, and we’re only truncating the top 47.5% of the parabolic trajectory on average, and we have added more ways for the room to interconvert vertical and horizontal moment.
Obviously I haven’t done any simulations or written down any equations to estimate the actual new distribution quantitatively. That would depend on the specific roof shape in ways I can’t easily capture in a simple equation (maybe someone else could, but I can’t). Even still, rephrased the way I put it above, that’s not nearly as surprising as it sounds when you stay vague and handwavy about it.
I suspect you could define a roof shape and a distribution of horizontal momentum vectors such that the balls would on average be deflected down faster than in the case of the flat roof.
Now, if I were to make the roof sticky instead of jagged, then sure, the balls spend more time there right at height h. But then the roof is absorbing the momentum and kinetic energy, producing heat in the process.