So I freely admit to not being a mathematician, however, I am a person who used to jump off swings a LOT, and indeed used to have informal contests with my friends about who could make the most reckless and dangerously long jumps from a swing with little thought to life or limb (we were teenage boys).
So having said that, I’m not sure how the OP calculated the results but 2m is far too small for a max distance. When we used to jump from the swing our goal was to always jump so far we made it out of the “swing area” where there was wood chippings on the ground and onto the grass beyond. Making to the grass was our bare minimum for a “good jump”
So I would say the minimum for a “good jump” us would be 3 meters, and anything above that would be reaching the standards of recklessness which got us a good teenage buzz going on.
AFAICT, the model used doesn’t ever allow the jumper to push against the ground. Since it also limits the time spent getting up to speed, that limits the amount of energy one can load into the system.
They also use l1 (distance between top of swing and seat) = 2m. I think that’s way too short for a swing. The modeled height of the seat from the ground is also small, at 40cm.
The final expression isn’t simple, but even keeping that height the same, the first term is l1 sin Φ, and that grows with l1. I didn’t check, but suspect v also grows if l1 increases. If so, the other terms also go up if l1 increases.
They also don’t model what Olympic swingers would do. I expect they would start by standing on the swing, grabbing the ropes high, and pulling hard, bringing their lower body almost horizontal, at a height at least a meter above the starting position, and then almost let go of the ropes, dropping down on the falling seat.
Just before jump-off, they might even try to hang below the seat as far as possible, like they do in ring or on the horizontal bar in gymnastics.
Notice that the amount of time you can spend on the swing is limited to 20 seconds. Of course with more time you can achieve much higher distances. The analysis is focused on small times/small angles approximation
This assumes the angle from the vertical is 'small' enough to simplify some trig terms. I'm pretty sure a swing can hit 90 degrees from vertical and have some doubts about that being small in the cancelling out trigonometric terms sense.
Yes, you're right, I made some approximations. The analysis is focused in the range of small \phi angles. However, it would be interesting to analyze what happens when \phi gets big. To do so you'll need to solve -probably numerically- the Lagrange equation for \phi (it's the second equation in the post), and then plug the solution into the distance equation.
As an aside: I'm not exactly a mechanical engineer, but based on the diagram at the beginning, I think there's an aspect of a swing-set that the author's not considering (although maybe it's mentioned in some of the papers referenced at the bottom)- namely, the fact that the seat is suspended by a _chain_, which you can pull or push to affect your motion. As far as I can tell, this article assumes a _rigid_ swing system.
I don't have the physics chops to work out how that would affect the results this article worked out, but I have to imagine it'd make it possible to get the swing going faster, sooner.
I was confused with the traveled distance vs jumping time plot.
Intuitively there should be three points at which the traveled distance is zero: at the two max swing heights, and at the min swing height (assuming your legs touch the ground at min swing height).
However the plot seems to only show zero traveled distances at min swing height? Or did I misunderstand the plot?
When you jump at the max swing heights you travel l*sin(phi). At these points you jump with zero velocity, however you've already moved from the starting position (phi=0)
My favorite is the vertical swing that can be found in some playgrounds. Lean forward and backwards to accelerate. It seems you can accelerate to infinite speed on these, and once you start to feel uncomfortable your natural instinct is to lean towards the center, which will only increase the speed, like in a pirouette.
Do you have a photo example by any chance? I can't quite picture what you mean and searching for vertical swing doesn't really show anything. I'm curious what you have in mind.
Oh right, thanks! I never tried one of those. They look kind of terrifying in a things can get out of hand spinning kind of way. Not that it would have bothered me as a kid
yeh, and ball bearings today are very good so little energy is lost due to friction. I often think there must be a way to abuse the swing mechanics to create a "engine" of sort that can defy time/gravity, or at least make a "potato cannon"
This is probably because the distance floor-swing is zero, which is not true in real life. I'll try to redo the maths with a non-zero distance and update the results :)
EDIT: I just added a new parameter to account for the distance from swing to floor, and now the maximum distance is 2 meters.
Two mistakes. One, 20s is way too short, you should be looking at 1 minute.
Also. T0=0 is fucking bullshit. Nobody trying to impress their fellow delinquents with height or distance is going to start with no motion. They’re going to fling themselves at the swing as hard as they possibly can. T0 is going to be over 10kph.
Why do you think 20s is a mistake? I mean, of course we can analyze what happens at longer time ranges, but I wouldn't say that was a mistake. Regarding the initial position we can also analyze what happens when some offset is allowed, but the game I designed consists on starting completely vertical.
In summary I agree it make sense to analyze your scenarios, buy I don't think they're mistakes.
This is probably because I've set the distance floor-swing to be zero, which is not true in real life. I'll try to redo the maths with a non-zero distance and update the results :)
EDIT: I just added a new parameter to account for the distance from swing to floor, and now the maximum distance is 2 meters.
Wouldn't the height of the swingset be a factor? I imagine if the swing is suspended from 50m you can continue winding up for a long time and jump from a height of 15+m which would give you a lot more horizontal movement before you hit the ground.
I mean the article doesn't say anything about calculating the farthest survivable jump from a swingset
So having said that, I’m not sure how the OP calculated the results but 2m is far too small for a max distance. When we used to jump from the swing our goal was to always jump so far we made it out of the “swing area” where there was wood chippings on the ground and onto the grass beyond. Making to the grass was our bare minimum for a “good jump”
So I would say the minimum for a “good jump” us would be 3 meters, and anything above that would be reaching the standards of recklessness which got us a good teenage buzz going on.