Hakugei, A Physics Meta, or: the physics of backspin on tennis balls
Back in the day when I read Tenipuri for the very first time, I possessed a combination of three unfortunate traits: 1. I was a high school student, 2. I was a physics nerd, and 3. I played tennis. All this means I ended up loving Fuji Shuusuke far more than I should, because his counters are a thing of beauty and look like something that could just pop up in my physics exam. I may or may not have spent weeks playing with ball spins and getting yelled at by my tennis coach trying to figure out Fuji's counters. I admit to nothing.
So now I will hold a little physics class on one of my favorite tenipuri shots. Several disclaimers before we start:
Bear in mind that I am in no way an expert in physics or tennis; do not read this meta for anything other than your own amusement, or the Gods of Tennis, Academia, and References will judge you.
While I will be talking about how certain tenipuri shots are physically possible, I will not be talking about whether or not a human being can hit these shots with a racket. That would require calculating force and rotation speed, and I’m not doing that because I hate math.
For ease of understanding, I will be oversimplifying certain things. That said, if you spot a mistake, please tell me so we can discuss how to fix it.
Considering the age of this fandom, I’m assuming most of you have learned about force and momentum in high school. If you haven’t, however, here is a short crash course:
Force is something that causes an object to move faster or slower in a certain direction. For example, when you hit a still tennis ball with a racket, your racket is applying force to the tennis ball.
Force is defined as Mass x Acceleration.
Momentum is what an object has when it’s moving. For example, when your tennis ball is flying over the net, your tennis ball has a forwards momentum.
Momentum is defined as Mass x Velocity
Applying force to an object causes its momentum to change. The longer force is applied, the more its momentum changes. For example, the longer you spend pushing a shopping cart, the faster it will roll after you let go of it.
Momentum is defined as Force x length of Time force is applied
Okay! Now we can start with the meta. The first shot I will be talking about is my favorite Fuji counter, Hakugei. @sandreeen, this meta is dedicated for you.
Amongst all of Fuji's counters, I think Hakugei is actually the simplest, and easiest to replicate by hand. Below is an illustration of Hakugei, plus the explanation we got when Fuji used it against Akugatawa Jirou for the first time.
From this illustration we can summarize the movements of the ball as follows:
1. The ball was hit with a slice, resulting in a backspin.
2. The ball then flies in an upwards curve.
3. The ball falls back down.
4. The ball bounces back towards Fuji.
5. The ball is caught by Fuji.
Now let's analyze these movements one by one.
1. The ball was hit with a slice, resulting in a backspin.
A backspin is simply a ball that rotates backwards towards the player. Fuji slices the bottom of the ball to create this spin. It could be illustrated as follows.
For the rest of the meta, all illustrations will be drawn with the assumption that Fuji is standing to your left, hitting the ball towards your right.
2. The ball flies in an upwards curve
Have you ever seen a fast-moving train pass by a pile of leaves? If you have, you may see the leaves get sucked into the path of the train. This is a demonstration of the Bernoulli Principle. The fast-moving train drags the air around it along, causing the air to move very fast. As a result, the air pressure near the train drops. A bit farther away, though, the air is still, meaning the air pressure there is higher. The pressure difference causes air from farther away to rush near the train, taking the leaves along.
This is basically what happens to Fuji’s Hakugei. As the tennis ball moves rightwards with a momentum of p, the air around it moves leftwards (A). Above the ball, the ball’s backspin (LA) drags the air to the left, moving it faster (A + LA). However, below the ball, the air was dragged to the right (LB), so it moves slower (A - LB). As the air above the ball is moving faster (A +LA) than the air below the ball (A - LB), the air pressure above the ball is lower than beneath the ball. Thus, the ball is rushed upwards (FM). This is called the Magnus Effect.
3. The ball falls down.
Not much to say here. Because of air resistance and drag and all those pesky forces that we generally ignore in high school physics, the ball will eventually slow down, meaning the effect of the ball’s spin on air speed will fade. Therefore, the Magnus effect will grow weaker. Gravity will eventually prevail, and the ball will come crashing back to the ground.
4. The ball bounces back towards Fuji
Now this is the interesting part--how the ball bounces back. Note that for this part, we will assume that the effects of air resistance and drag is negligible, and thus can be wholly ignored. Now, let’s first examine the most important forces acting on the ball in the instant before it hits the ground.
On the vertical axis, the ball is accelerating down, thanks to the force of gravity. This force is represented by Fg.
On the horizontal axis, the ball is moving to the right. The ball is not being affected horizontally by any kind of outside force, therefore, it moves at a constant speed. We can say that the ball is moving rightwards with a momentum of p.
The combination of Fg and p causes the ball to move in the direction of A.
Speaking of movement, other than vertical and horizontal movements, the ball is also spinning. Again, the ball’s spin is not being affected by any outside force. We can say that the ball is spinning constantly with an angular momentum of L.
And then the ball hits the ground.
Newton’s third law states that for every action, there is an equal and opposite reaction. On the vertical axis, we have Fg, which applies a downwards force upon the court. The court gives an equal and opposite reaction in Fn, which applies an upwards force upon the ball. This explains why the ball bounces upwards [1].
Meanwhile, in the horizontal axis, things get a little more complicated. We have the ball’s backwards spin interacting with the ground here. The ball’s spin produces an angular momentum of L, which gives a rightwards linear momentum of ps for the court.
Since the court is a stationary object and (hopefully) cannot move, the court’s momentum itself should stay at zero. We can now forget about the court. Any change of momentum will happen to the tennis ball, so let’s get back to that green fuzzy thing.
Newton’s third law states that for every action, there is an equal and opposite reaction. The Momentum Conservation Principle, which is derived from this law, states that in a collision, the change in the colliding objects’ momentum are also equal in magnitude, but opposite in directions.
According to the Momentum Conservation Principle, because its initial momentum (ps) is directed rightwards, after the collision, the ball’s momentum should change leftwards in the opposite direction. This change in momentum is represented by pr.
So when the ball hits the ground, horizontally, there are two different momentums fighting for dominance--pr and p [2]. To ensure the ball bounces to the left as Hakugei would, pr has to win this battle. Since pr is derived from ps, which in turn is derived from the ball’s angular momentum (L), the key to making Hakugei bounce back is to ensure that the ball’s angular momentum (L) is great enough to overpower its linear momentum (p). In other words, the ball has to spin quickly enough to overpower its own linear momentum (p).
Should pr be able to overpower p, the next direction of the tennis ball can be summed up by the following:
As you see, the ball will bounce back in the direction of B. No physics-defying magic necessary.
And now, let’s ask the real question: how great is the spin you need for the ball to bounce back? Well, not so great that you can’t achieve it by hand [3]. In fact, this was my favorite trick to do while running laps; the ball bounces back to me, so it’s easier to control, and I can maintain a slower pace without having to chase after recalcitrant balls. Of course, this only works on even grounds that can provide enough friction for the ball, such as tennis courts (if you’re using tennis balls) or basketball courts (if you’re using basketballs). On uneven grounds, the bounce is much less predictable, so beware where you try this trick!
5. The ball is caught by Fuji
To complete Hakugei, the ball must bounce back to the opposite side of the court and is caught by Fuji. Now considering that the ball has lost a lot of horizontal momentum when it hit the ground (in the battle between pr and p), it’s hard to believe that Hakugei could have enough force to return to Fuji all by itself. Therefore, Fuji cannot rely on the tennis ball itself to come back to him. Instead, he has to rely on another force: the wind.
How strong does the wind need to be to carry Hakugei all the way back to Fuji? Well, I honestly have no idea. Tennis balls are unexpectedly heavy, so I think it must be pretty strong, but I can’t say if it could reliably happen in your average tennis match. So, even though Hakugei should be physically possible, the wind factor makes it hard to use in an actual match.
In conclusion, ladies and gentlemen, this is why Hakugei is not only totally badass but also obeys the law of physics. If you read all the way here, thank you; you’re brilliant, and I hope my explanations didn’t give you too much trouble. I have no idea why I was insane enough to write this. However, I hope now you appreciate Fuji’s Hakugei as much as I do. After all, it takes a certain kind of person to be a Badass Normal in a world of Ten’imuhou no Kiwamis, Devil Modes, and Pirates of the World--and Fuji Shuusuke may just be the right genius to pull it off.
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[1] The reality is more complex than that, of course, involving air pressure inside the ball and such, but it’s not quite necessary to discuss that to understand Hakugei.
[2] There are a lot of other things going on here, especially with friction, which helps turns the ball’s point of contact with the court into a ‘pivot’ for pr’s momentum. There are also things going on with the air pressure inside the ball, which affects the length of time the ball stays in contact with the court and may further help pr overpower p.
[3] Theoretically, hitting a Hakugei with a racket should require a greater spin than throwing a backspin ball by hand. This is because the ball needs to travel further from the racket, against a strong headwind, not to mention other things that will reduce its spin such as air resistance.
[4] Hakuryuu probably works with similar principles, but with an angled backspin/topspin, depending on the situation. If I had to guess, I’d say Twist Serve works the same way, too. However, I still reserve judgement on whether Hakuryuu is physically possible.
