You can't bounce a ball under a table

- [Narrator] It's weirdly difficult to pass a bouncy ball under a table. The ball always wants to bounce out the same way it came in, which is odd. This is something that my friend Hugh Hunt showed me ages ago, and if you watch my channel a lot, you might be thinking, "Hey, that's really similar to the golf ball paradox from a few videos ago." If so, you are a lot smarter than I am because I made the whole golf ball paradox video without once thinking about balls bouncing under a table. But then

someone sent me this video. Here, the ball bounces around inside a square container and comes back out the way it came in without touching the back wall. AT who sent me the video felt that it was linked to the golf ball paradox. Only then did I think, "Well, hold on. If you can reduce the golf ball paradox down to a ball bouncing around the inside of a four-sided shape, why not go further? Why not reduce the problem to a ball bouncing between just two surfaces like Hugh showed me all those year

s ago?" And this is great because I gave an explanation for the golf ball paradox, but I wasn't that happy with it, and I feel like now I have a great opportunity to explain what's going on in a really intuitive way. So here's the plan. If I can show that a ball bouncing back to you from under a table makes intuitive sense, and if I can also show that that continues to make sense as we increase the number of surfaces that the ball bounces off, well then we just keep increasing the number of surf

aces until we have essentially a smooth surface. We have the cylinder from the golf ball paradox. In other words, we're taking the dynamics of a ball continuously rolling around the inside of a container and we are discretizing it into lots of little collisions. So let's first have a look at a ball bouncing under a table and see if we can make sense of that. So we'll assume the ball isn't spinning at all when it first hits the ground, but when it does hit the ground, the ground imparts some spin

onto the ball, which makes intuitive sense. And actually it's something you are familiar with from everyday life like the second bounce of a bouncy ball is often weird and unexpected. That's just because of the spin that was imparted on the ball from the first bounce. But to hammer the point home, imagine the interaction from the point of view of the ball from the ball's reference frame if you like. If you have a stationary ball and a wall rushes up to it and swipes it, the ball will obviously

start spinning. But look what happens now when the spinning ball hits the underside of the table. Well, when the ball hits the underside of the table, it's got backspin. And again, you know from experience what happens when a ball has backspin and it hits a surface, it comes back in the direction of the spin and that makes intuitive sense if you think about it in slow motion. Let's all pretend that I've keyed out my hand in this shot and you can only see the ball. So you've got a ball approachin

g a surface with some spin. The ball has angular momentum. In other words, it wants to keep spinning. Yes, I'm anthropomorphizing the ball, but that's okay. It will want to keep spinning when it touches the surface, but because of friction, it will have to roll during the collision. As a result, the ball ends up traveling back the way it came. In other words, the angular momentum of the ball is turned into a change in linear momentum. And in fact, the swipe from the underside of the table result

s in the ball spinning slowly in the other direction. The final bounce is pretty boring and what you'd expect, but it's the bounce that takes the ball back out again. Now let's double the number of faces the ball has to bounce around. We're now back to the square container or as I like to call it, a very low poly cylinder. I'm filming it from the side instead of straight down and I've made one of the walls transparent. I feel like this way it's easier to see what's going on with the spin of the

ball. So when the ball first hits the bottom plate, it gains some spin. Hopefully it's intuitive to see that the axis of that spin will be in the horizontal plane and perpendicular to the direction the ball is traveling. Here I've marked that axis of spin with a couple of pins. So what happens when the ball hits the second wall? Well, hopefully you can see that the patch of ball that touches the wall is moving downwards relative to the wall. So due to friction, the ball will roll upwards. In oth

er words, as a result of the spin, the ball will travel more steeply upwards than it would otherwise. But we should also expect the spin of the ball to change as a result of the interaction with that surface. To figure out what the new spin will be, think about what would happen to the ball if it didn't have any spin as it came into this second collision. Well, again, hopefully it's intuitive to see that the spin it would have after this collision would have an axis like this, but because it did

have spin coming into that collision, we should expect the new spin to be somewhere between the two, somewhere between this axis and the axis it had on the way in. So maybe it's spinning like this, and actually that's what we see on the slow motion video. So what will a ball with this spin traveling in this direction do when it touches the top surface? Well, let's imagine that the top surface is transparent as well. So look, you can see that the patch of ball that touches the top surface is tra

veling in this direction. And again, because it's backspin, because of friction, that will impart momentum onto the ball in the opposite direction. And in this case, it's enough to switch from the ball traveling into the container, to the ball traveling out of the container. And again, the spin of the ball will change. It's somewhere between this spin and the spin that it had coming in. And again, that's what we see in the slow motion footage. You can carry on reasoning in this way until the bal

l leaves the container. What if we double the number of faces again? So it's an eight-sided container. By the way, if anyone feels like actually building an octagonal container to test this thing, then please do share the results in the Science Fair project channel on my Discord server. Link to that in the description. But I'll be using animation for now, and you'll notice that with more faces, each collision with a face has less of an effect on the trajectory of the ball. That's for two reasons

. First, the ball is hitting the next face at a shallower angle. It's less of a head-on collision. And secondly, because the ball is already spinning from the first collision, the movement of the surface of the ball underneath the surface of the container, well, they almost match, almost but not exactly. Let's actually have a look at what happens between two successive bounces when you've got this shallow angle between the faces. Just like with the square container, the spin of the ball modifies

the final trajectory of the ball as it leaves the second surface, just less so. And just like with the square container, the new spin axis of the ball will be somewhere between this axis and the axis coming in. Those two axes are much closer to each other, so the change of the spin axis is less as well. So if you have smaller changes per bounce, but more bounces, you end up with the same result. The ball comes out after about one full turn. I say about one full turn because you might remember f

rom the Golf Ball Paradox video that the square root of seven divided by two gives you the number of rotations around the container for every full vertical oscillation within the container. But of course for the ball to get outta the container, that's just half a vertical oscillation. So the number of rotations before the ball comes out of the container, we should expect to be the square root of seven divided divided by two, divided by two and that's a little less than one. And crucially, there'

s always that inflection point where the ball transitions from traveling into the container to traveling out of the container. We saw that with the square-shaped container. It's when the ball hits the top wall that it transitions from going in to going out. In a hexagonal container, we'd expect that inflection to happen about halfway round. But the same principle applies. It's the point when the angular momentum of the ball is pointing in the right direction to cause the ball to switch direction

s. If we double the number of faces to 16, then it's starting to look a lot like a cylinder. In other words, the golf ball paradox is just what happens in the limit as the number of faces of your container approaches infinity. In the original video I said that if you assume the axis of rotation of the ball is fixed as if it were a gyroscope, then we would expect the ball to oscillate up and down, but that it would make one vertical oscillation after just one single turn around the cylinder. I sa

id that the reason it takes more than one turn is because the cylinder kind of drags the axis of spin of the ball around as opposed to the axis being fixed. And I feel like that's easier to see in the discretized version. When the ball bounces off a surface with spin, it forces the ball to change direction, but the collision also changes the spin of the ball. We see that very nicely in the slow-mo footage here. And of course, as the number of faces of the container tends to towards infinity, the

number of collisions tends towards infinity, and those discrete changes in the direction of the spin of the ball becomes a continuous change in the direction of the spin of the ball as it rolls around the inside of the cylinder. A couple more really interesting things that came up after I published the Golf Ball Paradox video. Patrick Deful emailed me to say he created a simulation of the golf ball paradox. He'd also previously created simulations of the ball on a turntable. I'll leave a link t

o all of those in the description. It really is nice to see that it works in simulation as well. One final thought, discretizing the problem in this way is really interesting for two reasons. First, it's what Newton did when he was first looking at orbits. Well, maybe it was hook actually. But anyway, because orbital mechanics have nonlinear equations, it's hard to extrapolate where a planet will be in the future. But if you imagine it instead as gravity in parting discrete impulses onto massive

objects, you can go step by step to figure out where a planet will be in the future. On that basis, Newton was able to prove Kepler's equal area law. And interestingly, software that simulates the motion of charged particles uses the same method. There's something really interesting about the business of spam. A lot of the spam we get is sent illegally, but a lot of it is completely legal, including some of the spam phone calls that we get. But despite being legal, they're actually potentially

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