[0:00] What happens when you combine a really
[0:02] bouncy ball with a really bouncy
[0:04] surface?
[0:05] It stops dead. But look, if I tweak it
[0:08] slightly,
[0:09] now it's a really good bounce. This is
[0:11] when I realized I don't really
[0:13] understand bouncing. Because it turns
[0:16] out if you want to maximize bounce, it's
[0:18] not enough to just combine two really
[0:21] bouncy things. But what I found really
[0:23] surprising was that when I tried to find
[0:26] the optimum bounce using math, I ended
[0:29] up generating fractals. First, I want to
[0:32] be clear about what we're doing in this
[0:33] video. Like I've shown you really bouncy
[0:36] things before, like the atomic
[0:37] trampoline and well, maybe you've seen
[0:39] an advert for the world's bounciest ball
[0:43] or whatever, but that type of
[0:44] optimization isn't what we're dealing
[0:46] with here. Like those examples are about
[0:49] avoiding energy loss during the bounce.
[0:52] So, the atomic trampoline minimizes
[0:55] plastic deformation, the super bouncy
[0:58] balls minimize internal friction during
[1:01] the collision. But the thing is, this
[1:03] bouncy ball and this bouncy surface are
[1:06] both really good at retaining energy.
[1:08] So,
[1:09] what went wrong here? Well, look at this
[1:12] ball bearing on this rubber sheet. The
[1:15] ball is really hard and the surface is
[1:18] really flexible and that gives you a
[1:20] really good bounce. And with the super
[1:22] bouncy ball, the ground is really hard,
[1:25] but the ball is really flexible and that
[1:27] gives you a good bounce as well. The
[1:30] problem seems to arise when you have a
[1:32] ball and a surface that are similarly
[1:35] flexible. For example, Orbeez are really
[1:37] flexible and so is this rubber sheet.
[1:40] And right now, the combination is super
[1:42] bouncy. But if I add a little bit of
[1:44] weight to the rubber sheet, well,
[1:46] suddenly it all goes wrong. And that's
[1:48] why I built this contraption to try and
[1:50] figure out what's going on. See, it's
[1:52] when I change the weight that it goes
[1:55] from a good bounce to a bad bounce. It's
[1:59] weird, isn't it? Actually, this
[2:00] particular configuration is quite fun
[2:02] because when gravity pulls the ball back
[2:04] down, it gets a kind of second kick,
[2:08] which is such an odd behavior. By the
[2:10] way, this is somewhat related to how
[2:11] children typically injure themselves on
[2:14] a trampoline. A child can often get an
[2:15] unexpected second kick if there are
[2:18] other people on the trampoline. Before I
[2:20] show you what I figured out with this
[2:21] device, I also had a good chat with the
[2:23] professor of the author of this paper.
[2:26] In the paper, they're thinking about it
[2:27] in terms of a golf ball and a golf club.
[2:30] And basically, they do a whole lot of
[2:31] simulations where they simplify it down
[2:33] to four things: the mass of the ball,
[2:35] the stiffness of the ball, the mass of
[2:37] the golf club, and the stiffness of the
[2:39] golf club. And when I say stiffness, I
[2:41] mean in the spring sense. So, like, if a
[2:44] spring is hard to stretch, then it's a
[2:47] stiff spring. If it's easy to stretch,
[2:49] then it's a less stiff spring. And a
[2:51] golf ball and a golf club are in some
[2:53] sense springy. So, we talk about
[2:56] stiffness. Anyway, when they run a whole
[2:57] lot of simulations where they kept three
[2:59] things constant and varied the fourth
[3:02] thing, they discovered something really
[3:03] surprising. See, in this graph, they're
[3:05] varying the mass of the golf club while
[3:08] keeping everything else fixed. And the
[3:10] height of the curve is how good the
[3:13] bounce was. Specifically, it's the
[3:15] velocity of the ball when it leaves the
[3:17] club. And you can see how the quality of
[3:19] the bounce well, it goes up and down as
[3:22] the mass varies. So, you get an optimal
[3:24] bounce for these masses. But, if you
[3:27] change the mass of the ball to one of
[3:29] these masses, well, you'd actually get a
[3:31] terrible bounce. To see what's going on,
[3:33] I built this simulation that represents
[3:36] the model that they were using in the
[3:38] paper, and it means we can see actually
[3:41] what's physically going on with
[3:43] different examples. So, this is the mass
[3:45] of the ball. This spring represents the
[3:47] stiffness of the ball. This is the mass
[3:49] of the club. And this spring represents
[3:51] the stiffness of the club. Let's see
[3:53] what happens if we set the mass of the
[3:57] club to be about twice the mass of the
[3:59] ball. So, the ball engages with the
[4:01] club, it starts to squish, but also the
[4:04] club is being pushed now, so that starts
[4:06] to squish as well. Crucially, look, the
[4:08] ball disengages from the club when the
[4:11] club is at almost maximum squish, but
[4:13] keep watching because but maybe the club
[4:16] will catch up with the ball and it'll
[4:17] give it a second kick.
[4:23] No, it just missed.
[4:25] Crucially though, look, see how much
[4:27] vibrational energy there is in the club.
[4:30] That vibration is energy locked away in
[4:33] the club that could have been given to
[4:36] the ball but wasn't. So, this represents
[4:39] a bad bounce. Let's compare that to when
[4:42] the mass of the club is about 2/3 of the
[4:45] mass of the ball. So, again, the ball
[4:47] starts to squish, and then the club
[4:50] starts to squish as well. Right, the
[4:52] club is now at maximum squish. The ball
[4:54] didn't disengage this time because the
[4:57] spring is still compressed. So, as the
[4:59] club spring is re-expanding, it's
[5:01] actually working hard against the mass
[5:04] of the ball because the spring is still
[5:06] compressed, and that's slowing it down
[5:09] at just the right rate so that when the
[5:11] ball finally disengages, look, the club
[5:13] mass is at rest at its rest length. That
[5:17] means there's no vibrational energy left
[5:19] in the club. So, all of the energy went
[5:22] into the kinetic energy of the ball.
[5:24] This is a perfect bounce. But if you
[5:26] look at the graph in the paper, you can
[5:27] see that actually there are lots of
[5:30] points of maximum bounce. So, let's also
[5:33] look at when the club mass is about 1/6
[5:37] of the mass of the ball. So, really it's
[5:39] the same situation except if you watch
[5:41] the club mass, actually it goes through
[5:43] multiple oscillations before that
[5:46] perfect separation that we saw before.
[5:49] And it turns out that when you do the
[5:50] math, you get quite a simple result that
[5:53] actually, hopefully, makes some
[5:56] intuitive sense. So, it's all about the
[5:59] natural oscillation of the club. See,
[6:01] the club naturally oscillates like this
[6:03] over time. But because it's being pushed
[6:06] by the ball throughout the whole
[6:08] collision, this graph of position gets
[6:10] skewed like this. Now, what I'm showing
[6:12] you is a perfect bounce. Because, look,
[6:15] the club is back at its rest position at
[6:18] the end of the collision. That tells us
[6:20] that there's no energy stored in the
[6:21] spring at the end of the collision. But
[6:23] also notice that the graph of position
[6:26] is flat at the end. So, we know there's
[6:29] no kinetic energy in the club mass,
[6:32] either. Now, you'll notice that the club
[6:34] mass has gone through two and a half
[6:36] oscillations. But let's stiffen the
[6:39] spring so it oscillates faster.
[6:41] It's still a perfect bounce, but now
[6:43] it's gone through three and a half
[6:45] oscillations.
[6:47] And there's a perfect bounce at four and
[6:49] a half oscillations, and so on. And
[6:51] look, if you go through a whole number
[6:53] of oscillations, that's when you get a
[6:54] perfectly bad bounce. How does that
[6:57] compare to the natural oscillation of
[6:58] the ball? Well, the ball always goes
[7:01] through exactly half an oscillation
[7:03] during the collision because
[7:05] well, it's not sticky. So, here's the
[7:07] simple rule. The club needs to go
[7:09] through a whole number of oscillations
[7:11] plus a half in the time it takes the
[7:13] ball to go through half an oscillation
[7:16] if you want to get a good bounce. If you
[7:18] want a bad bounce, just go for a whole
[7:20] number of oscillations. In general,
[7:22] though, the surface needs to jiggle
[7:23] faster than the ball for interesting
[7:26] things to happen. But in reality, it's
[7:28] hard to find a combination of ball and
[7:31] surface where the surface vibrates
[7:33] faster than the ball, but not loads
[7:36] faster. In my setup, the ball always
[7:38] vibrates faster than the surface.
[7:40] Doesn't matter what I tried in terms of
[7:42] springs and everything, but clearly
[7:44] interesting things still happen. I think
[7:46] that's because we made some simplifying
[7:49] assumptions, but actually there is
[7:50] another issue. See, in the simulation
[7:53] some energy can be left behind in the
[7:55] oscillating surface as we've seen, but
[7:57] you can see in this footage that you can
[7:59] also end up with oscillation energy left
[8:02] in the ball. But actually because of the
[8:04] way this simulation is put together,
[8:06] there's no way to account for
[8:08] oscillation energy left behind in the
[8:10] ball because the spring doesn't have any
[8:12] mass, so it can only oscillate when it's
[8:15] in contact with another mass. In other
[8:17] words, during the collision. So I
[8:19] decided to make my own simulation. In
[8:22] this version, the mass of the ball is
[8:25] split in two. So the ball can now
[8:29] oscillate on its own. Honestly, I'm not
[8:31] sure whether it's a reasonable model of
[8:34] reality, but I just want to show you
[8:36] what happens when you explore it because
[8:39] some of it's really interesting. So
[8:40] yeah, I can vary the stiffness of the
[8:43] springs and the masses or I can sweep
[8:47] through one of those values to get a
[8:49] plot. So look,
[8:51] you can see here this is a lot like the
[8:53] graph from the paper. So look, I can
[8:55] select a maximum and then play that.
[9:02] Oh. [sighs]
[9:03] That's satisfying, isn't it? Let's look
[9:05] at a bad one. Actually, look at the
[9:06] worst one over here.
[9:10] Oh, that's awful.
[9:13] In the plot, you can also see the number
[9:15] of collisions. So it's interesting,
[9:16] isn't it? The collisions go up and up.
[9:18] This is the internal vibrational energy
[9:21] of the ball after the collision. It
[9:23] doesn't actually get that high. So what
[9:25] we can do is sweep two things at once if
[9:28] we want to try and say optimize the
[9:30] amount of jiggle in the ball after the
[9:32] collision. We'll vary the mass of the
[9:33] golf club and the spring constant of the
[9:37] golf club. So, here's a heat map showing
[9:39] how good the bounce is. If I go here,
[9:42] this is a good bounce. See what that
[9:43] looks like.
[9:47] That's getting it. And then here's a
[9:49] really bad bounce, let's say.
[9:54] And if we switch to the internal energy,
[9:58] 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
[15:48] do that, may I humbly suggest the
[15:50] sponsor of this video, Henson Shaving.
[15:52] With a Henson AL13, the training wheels
[15:54] are off. You don't press hard. You can
[15:56] learn the shape of your face because you
[15:58] can actually feel it.
[16:00] And you end up with an actually
[16:02] enjoyable shave. No lubricating strip,
[16:05] no aftershave balm. The handle is more
[16:07] expensive, but you only ever buy it
[16:10] once. And the blades are pennies. So, it
[16:14] actually ends up being cheaper after
[16:15] just a few months. Go to
[16:17] hensonshaving.com/steve
[16:18] mold and use code steve mold at checkout
[16:21] to get 100 free blades with your
[16:23] purchase of a Henson AL13. That's
[16:25] equivalent to about 3 or 4 years of
[16:27] irritation-free shaving. The link is
[16:29] also in the description, so check out
[16:31] Henson Shaving today. I hope you enjoyed
[16:33] this video. If you did, don't forget to
[16:35] hit subscribe. And the algorithm thinks
[16:37] you'll enjoy this video next.