The Anti Trampoline Effect

[00:00] What happens when you combine a really

[00:02] bouncy ball with a really bouncy

[00:04] surface?

[00:05] It stops dead. But look, if I tweak it

[00:08] slightly,

[00:09] now it's a really good bounce. This is

[00:11] when I realized I don't really

[00:13] understand bouncing. Because it turns

[00:16] out if you want to maximize bounce, it's

[00:18] not enough to just combine two really

[00:21] bouncy things. But what I found really

[00:23] surprising was that when I tried to find

[00:26] the optimum bounce using math, I ended

[00:29] up generating fractals. First, I want to

[00:32] be clear about what we're doing in this

[00:33] video. Like I've shown you really bouncy

[00:36] things before, like the atomic

[00:37] trampoline and well, maybe you've seen

[00:39] an advert for the world's bounciest ball

[00:43] or whatever, but that type of

[00:44] optimization isn't what we're dealing

[00:46] with here. Like those examples are about

[00:49] avoiding energy loss during the bounce.

[00:52] So, the atomic trampoline minimizes

[00:55] plastic deformation, the super bouncy

[00:58] balls minimize internal friction during

[01:01] the collision. But the thing is, this

[01:03] bouncy ball and this bouncy surface are

[01:06] both really good at retaining energy.

[01:08] So,

[01:09] what went wrong here? Well, look at this

[01:12] ball bearing on this rubber sheet. The

[01:15] ball is really hard and the surface is

[01:18] really flexible and that gives you a

[01:20] really good bounce. And with the super

[01:22] bouncy ball, the ground is really hard,

[01:25] but the ball is really flexible and that

[01:27] gives you a good bounce as well. The

[01:30] problem seems to arise when you have a

[01:32] ball and a surface that are similarly

[01:35] flexible. For example, Orbeez are really

[01:37] flexible and so is this rubber sheet.

[01:40] And right now, the combination is super

[01:42] bouncy. But if I add a little bit of

[01:44] weight to the rubber sheet, well,

[01:46] suddenly it all goes wrong. And that's

[01:48] why I built this contraption to try and

[01:50] figure out what's going on. See, it's

[01:52] when I change the weight that it goes

[01:55] from a good bounce to a bad bounce. It's

[01:59] weird, isn't it? Actually, this

[02:00] particular configuration is quite fun

[02:02] because when gravity pulls the ball back

[02:04] down, it gets a kind of second kick,

[02:08] which is such an odd behavior. By the

[02:10] way, this is somewhat related to how

[02:11] children typically injure themselves on

[02:14] a trampoline. A child can often get an

[02:15] unexpected second kick if there are

[02:18] other people on the trampoline. Before I

[02:20] show you what I figured out with this

[02:21] device, I also had a good chat with the

[02:23] professor of the author of this paper.

[02:26] In the paper, they're thinking about it

[02:27] in terms of a golf ball and a golf club.

[02:30] And basically, they do a whole lot of

[02:31] simulations where they simplify it down

[02:33] to four things: the mass of the ball,

[02:35] the stiffness of the ball, the mass of

[02:37] the golf club, and the stiffness of the

[02:39] golf club. And when I say stiffness, I

[02:41] mean in the spring sense. So, like, if a

[02:44] spring is hard to stretch, then it's a

[02:47] stiff spring. If it's easy to stretch,

[02:49] then it's a less stiff spring. And a

[02:51] golf ball and a golf club are in some

[02:53] sense springy. So, we talk about

[02:56] stiffness. Anyway, when they run a whole

[02:57] lot of simulations where they kept three

[02:59] things constant and varied the fourth

[03:02] thing, they discovered something really

[03:03] surprising. See, in this graph, they're

[03:05] varying the mass of the golf club while

[03:08] keeping everything else fixed. And the

[03:10] height of the curve is how good the

[03:13] bounce was. Specifically, it's the

[03:15] velocity of the ball when it leaves the

[03:17] club. And you can see how the quality of

[03:19] the bounce well, it goes up and down as

[03:22] the mass varies. So, you get an optimal

[03:24] bounce for these masses. But, if you

[03:27] change the mass of the ball to one of

[03:29] these masses, well, you'd actually get a

[03:31] terrible bounce. To see what's going on,

[03:33] I built this simulation that represents

[03:36] the model that they were using in the

[03:38] paper, and it means we can see actually

[03:41] what's physically going on with

[03:43] different examples. So, this is the mass

[03:45] of the ball. This spring represents the

[03:47] stiffness of the ball. This is the mass

[03:49] of the club. And this spring represents

[03:51] the stiffness of the club. Let's see

[03:53] what happens if we set the mass of the

[03:57] club to be about twice the mass of the

[03:59] ball. So, the ball engages with the

[04:01] club, it starts to squish, but also the

[04:04] club is being pushed now, so that starts

[04:06] to squish as well. Crucially, look, the

[04:08] ball disengages from the club when the

[04:11] club is at almost maximum squish, but

[04:13] keep watching because but maybe the club

[04:16] will catch up with the ball and it'll

[04:17] give it a second kick.

[04:23] No, it just missed.

[04:25] Crucially though, look, see how much

[04:27] vibrational energy there is in the club.

[04:30] That vibration is energy locked away in

[04:33] the club that could have been given to

[04:36] the ball but wasn't. So, this represents

[04:39] a bad bounce. Let's compare that to when

[04:42] the mass of the club is about 2/3 of the

[04:45] mass of the ball. So, again, the ball

[04:47] starts to squish, and then the club

[04:50] starts to squish as well. Right, the

[04:52] club is now at maximum squish. The ball

[04:54] didn't disengage this time because the

[04:57] spring is still compressed. So, as the

[04:59] club spring is re-expanding, it's

[05:01] actually working hard against the mass

[05:04] of the ball because the spring is still

[05:06] compressed, and that's slowing it down

[05:09] at just the right rate so that when the

[05:11] ball finally disengages, look, the club

[05:13] mass is at rest at its rest length. That

[05:17] means there's no vibrational energy left

[05:19] in the club. So, all of the energy went

[05:22] into the kinetic energy of the ball.

[05:24] This is a perfect bounce. But if you

[05:26] look at the graph in the paper, you can

[05:27] see that actually there are lots of

[05:30] points of maximum bounce. So, let's also

[05:33] look at when the club mass is about 1/6

[05:37] of the mass of the ball. So, really it's

[05:39] the same situation except if you watch

[05:41] the club mass, actually it goes through

[05:43] multiple oscillations before that

[05:46] perfect separation that we saw before.

[05:49] And it turns out that when you do the

[05:50] math, you get quite a simple result that

[05:53] actually, hopefully, makes some

[05:56] intuitive sense. So, it's all about the

[05:59] natural oscillation of the club. See,

[06:01] the club naturally oscillates like this

[06:03] over time. But because it's being pushed

[06:06] by the ball throughout the whole

[06:08] collision, this graph of position gets

[06:10] skewed like this. Now, what I'm showing

[06:12] you is a perfect bounce. Because, look,

[06:15] the club is back at its rest position at

[06:18] the end of the collision. That tells us

[06:20] that there's no energy stored in the

[06:21] spring at the end of the collision. But

[06:23] also notice that the graph of position

[06:26] is flat at the end. So, we know there's

[06:29] no kinetic energy in the club mass,

[06:32] either. Now, you'll notice that the club

[06:34] mass has gone through two and a half

[06:36] oscillations. But let's stiffen the

[06:39] spring so it oscillates faster.

[06:41] It's still a perfect bounce, but now

[06:43] it's gone through three and a half

[06:45] oscillations.

[06:47] And there's a perfect bounce at four and

[06:49] a half oscillations, and so on. And

[06:51] look, if you go through a whole number

[06:53] of oscillations, that's when you get a

[06:54] perfectly bad bounce. How does that

[06:57] compare to the natural oscillation of

[06:58] the ball? Well, the ball always goes

[07:01] through exactly half an oscillation

[07:03] during the collision because

[07:05] well, it's not sticky. So, here's the

[07:07] simple rule. The club needs to go

[07:09] through a whole number of oscillations

[07:11] plus a half in the time it takes the

[07:13] ball to go through half an oscillation

[07:16] if you want to get a good bounce. If you

[07:18] want a bad bounce, just go for a whole

[07:20] number of oscillations. In general,

[07:22] though, the surface needs to jiggle

[07:23] faster than the ball for interesting

[07:26] things to happen. But in reality, it's

[07:28] hard to find a combination of ball and

[07:31] surface where the surface vibrates

[07:33] faster than the ball, but not loads

[07:36] faster. In my setup, the ball always

[07:38] vibrates faster than the surface.

[07:40] Doesn't matter what I tried in terms of

[07:42] springs and everything, but clearly

[07:44] interesting things still happen. I think

[07:46] that's because we made some simplifying

[07:49] assumptions, but actually there is

[07:50] another issue. See, in the simulation

[07:53] some energy can be left behind in the

[07:55] oscillating surface as we've seen, but

[07:57] you can see in this footage that you can

[07:59] also end up with oscillation energy left

[08:02] in the ball. But actually because of the

[08:04] way this simulation is put together,

[08:06] there's no way to account for

[08:08] oscillation energy left behind in the

[08:10] ball because the spring doesn't have any

[08:12] mass, so it can only oscillate when it's

[08:15] in contact with another mass. In other

[08:17] words, during the collision. So I

[08:19] decided to make my own simulation. In

[08:22] this version, the mass of the ball is

[08:25] split in two. So the ball can now

[08:29] oscillate on its own. Honestly, I'm not

[08:31] sure whether it's a reasonable model of

[08:34] reality, but I just want to show you

[08:36] what happens when you explore it because

[08:39] some of it's really interesting. So

[08:40] yeah, I can vary the stiffness of the

[08:43] springs and the masses or I can sweep

[08:47] through one of those values to get a

[08:49] plot. So look,

[08:51] you can see here this is a lot like the

[08:53] graph from the paper. So look, I can

[08:55] select a maximum and then play that.

[09:02] Oh. [sighs]

[09:03] That's satisfying, isn't it? Let's look

[09:05] at a bad one. Actually, look at the

[09:06] worst one over here.

[09:10] Oh, that's awful.

[09:13] In the plot, you can also see the number

[09:15] of collisions. So it's interesting,

[09:16] isn't it? The collisions go up and up.

[09:18] This is the internal vibrational energy

[09:21] of the ball after the collision. It

[09:23] doesn't actually get that high. So what

[09:25] we can do is sweep two things at once if

[09:28] we want to try and say optimize the

[09:30] amount of jiggle in the ball after the

[09:32] collision. We'll vary the mass of the

[09:33] golf club and the spring constant of the

[09:37] golf club. So, here's a heat map showing

[09:39] how good the bounce is. If I go here,

[09:42] this is a good bounce. See what that

[09:43] looks like.

[09:47] That's getting it. And then here's a

[09:49] really bad bounce, let's say.

[09:54] And if we switch to the internal energy,

[09:58] there's a high point right here. So,

[10:00] this is where we end up with lots of

[10:03] internal energy in the ball at the end.

[10:05] Look at that.

[10:07] That is a terrible bounce. Most of the

[10:09] energy is in the ball. But

[10:11] interestingly, there's some structure

[10:14] down here. Let's do a different sweep

[10:16] with the mass of the club and the

[10:19] stiffness of the club.

[10:21] Look at that. Almost like a fractal.

[10:23] Maybe it is a fractal. Like if I zoom in

[10:25] here,

[10:26] isn't that weird?

[10:28] But anyway, an unexpected fractal

[10:31] appears. Something else I discovered

[10:33] that matches real life. Like if there

[10:36] are several collisions within a bounce,

[10:38] are you only looking at what happens

[10:40] after the first collision, or do you

[10:42] count all the collisions? This is a

[10:43] perfect example.

[10:45] It stops dead. It's a terrible bounce.

[10:47] But then on the second kick, it's an

[10:48] amazing bounce. And I got something like

[10:50] this in my studio. Look at that. It's a

[10:53] terrible bounce. The ball stops dead,

[10:56] but moments later it gets a second kick.

[10:59] So, how does this compare to reality? To

[11:01] compare to the model, I don't need to

[11:02] worry about weighing things or

[11:05] calculating the stiffness, because I can

[11:07] just measure the period of oscillation

[11:10] from the video. Okay, so we're

[11:11] interested in how many times does the

[11:13] surface oscillate during half an

[11:15] oscillation of the ball. The model from

[11:17] the paper predicts good bounces at 1 and

[11:20] 1/2, 2 and 1/2, 3 and 1/2, and so on.

[11:23] But I was never able to get the surface

[11:24] to oscillate fast enough to explore

[11:27] that, though I did get quite close. In

[11:28] our data, the surface only ever goes

[11:30] through a fraction of an oscillation

[11:33] during the collision. This is where the

[11:34] model from the paper predicts the good

[11:36] and bad bounces will be, and these are

[11:37] our good and bad bounces marked in green

[11:40] and red. We never get into the

[11:41] interesting region because my surface is

[11:43] too slow. But clearly the data doesn't

[11:45] match what's going on at the boring end

[11:47] either. But here's the cool thing. In my

[11:49] version of the simulation, interesting

[11:50] things can happen when the surface

[11:53] oscillates slower than the ball. In

[11:55] fact, it's basically a mirror image when

[11:57] you look at it on a log scale so that

[11:59] you get bad bounces when the surface

[12:01] oscillates faster than the ball and good

[12:04] and bad bounces when the ball oscillates

[12:06] faster than the surface. In reality,

[12:08] it's more complicated than that for

[12:09] several reasons, and our data doesn't

[12:12] really fit this model either. But what's

[12:14] clear is the model is very spiky. Like,

[12:17] you don't have to change very much for

[12:19] the quality of the bounce to change

[12:21] drastically. And our data is also very

[12:24] spiky. So our data has roughly the right

[12:27] character, but I'm not sure we can claim

[12:29] much more than that. One cool thing

[12:30] about both of these models is they

[12:32] predict what should happen for a really

[12:35] bouncy ball on concrete or a steel ball

[12:38] bearing on a rubber sheet. See, this is

[12:40] a plot of how good the bounce is as the

[12:42] stiffness of the surface increases. And

[12:45] the line still goes up and down in that

[12:48] counterintuitive way, but as the

[12:50] stiffness increases, actually that

[12:52] effect gets less and less until it

[12:55] eventually just flattens out. In other

[12:57] words, if you've got a squishy ball and

[12:59] a concrete floor, you don't need to

[13:01] worry about comparing their periods of

[13:04] oscillation. And then you're back to

[13:06] just worrying about dissipation through

[13:09] internal friction and things like that.

[13:10] And the same is true if you've got a

[13:12] stiff ball and a squishy surface. So if

[13:15] you want to find the bounciest

[13:17] combination of ball and surface, you

[13:20] can't just combine the two bounciest

[13:23] things you own. You have to tune their

[13:25] vibrations as well. Otherwise, you could

[13:28] end up with a dud. I messed up while

[13:31] making this video and I want to share

[13:33] the story with you as a cautionary tale.

[13:35] So, I wanted this surface to vibrate

[13:37] faster than the ball vibrates. So, I end

[13:40] up buying like stiffer and stiffer

[13:42] springs, but nothing was working. The

[13:46] frequency of this thing barely shifted

[13:49] at all. It got to the point where I had

[13:50] to print this part in a stronger plastic

[13:54] and I was using these nuts to tension

[13:57] the springs, which just took ages. I

[14:01] eventually realized the problem. The

[14:03] stiffer springs were heavier and so

[14:05] adding mass to the oscillator, which was

[14:09] counteracting the additional stiffness

[14:12] of the oscillator. The thing that

[14:14] eventually worked was ungluing this part

[14:16] so I could slide it up and down and then

[14:18] clipping it in place at different spring

[14:21] extensions. But, the surprising thing is

[14:24] that this is an almost perfect analogy

[14:26] for cartridge razors. I want to be clear

[14:28] that's a segue, not a crowbar. The

[14:30] difference being that the link is

[14:31] actually really strong. It's a good

[14:33] link. It's a strong link. I mean, you be

[14:35] the judge. Okay, here's the thing. The

[14:37] problem with plastic cartridges is

[14:38] there's always going to be a little bit

[14:40] of give in the blade, so you get an

[14:42] inconsistent shave. To counteract that,

[14:44] manufacturers just add more blades in

[14:47] the hope of catching more hairs during

[14:50] each pass. But, the first blade will cut

[14:52] some hairs, so the subsequent blades

[14:55] will be scraping against skin in certain

[14:57] places leading to irritation. So, then

[15:00] they have to make the blades

[15:01] deliberately more springy, which makes

[15:03] the shave worse. So, they add even more

[15:05] blades. Gets to the point where there

[15:07] are so many blades, you have to add a

[15:09] lubricating strip to overcome the

[15:11] friction and people over time learn to

[15:13] press really hard and so you also have

[15:16] to add a post-shave balm and things like

[15:19] that. Complexity piled on top of

[15:20] complexity just because the

[15:22] manufacturers weren't able to implement

[15:25] the simple fix at the very beginning of

[15:28] just holding a single blade really

[15:31] family because you can never do that

[15:33] with plastic. The solution is to not buy

[15:36] cartridge razors. Instead, buy a

[15:38] precision engineered aluminum handle

[15:41] that holds a single blade firmly in

[15:44] exactly the right position at exactly

[15:46] the right angle. And if you're going to

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[16:33] this video. If you did, don't forget to

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