Game Theory - How to BREAK Mario! (Mario Tennis Aces)

Episode Date: October 9, 2023

Join Game Theory Host MatPat as he breaks down the science behind Mario Tennis Aces!

internet, welcome to game theory, spending more time analyzing fictional athletics than actually going outside. I know I recently said that Mario is classified as overweight according to the BMI scale, but one thing that BMI doesn't do a good job of factoring in is muscle mass, which is heavier than fat. And when you take a look at Mario, sure, he may appear like an adorable chubby plumber, but he in fact may be the most athletic character to ever appear in video game dumb. I mean, he's appeared in 25 sports spin-off titles, competed in every single Olympic event, both summer and winter, and mastered everything from golf and soccer,

to non-traditional activities like having Waluigi slowly grind up on him. Oh, Waluigi, you casual lover. You might not make the smash Bros cut, but you'll always have a place on Mario's dance card. You'll be the challenger approaching from the rear every time. Mario's latest venture is Mario Tennis Aces, on the Nintendo Switch. You see, during matches, players can fill up an energy gauge by rallying the ball. Then, with enough energy, you're able to hit a zone shot,

a fast-moving spite that's near impossible to return. Near impossible. You see, that's where there's this brilliant little bit of strategy involved. You have the choice to either hit your zone shot into the corner of the court, where your opponent will have no chance at returning it, or shooting it directly at them. If you go this route, sure, they have the possibility of returning the hit, but if they miss,

Their racket may just snap in half, forcing them to forfeit the match entirely. Apparently with Mario hoarding all the gold coins, no one can afford more than one racket in this universe. Anyway, this all got me thinking, is it actually possible to launch a tennis ball so hard that it can break an opponent's tennis racket? And if it can break a tennis racket, what other horrific things could that ball do? I know we've talked about how deadly some things in the Mario games can get, but could its deadliest weapon actually be a simple tennis ball? Let's pull out our Ti-83s and find out. As with all games, we'll need to determine a census scale first and foremost. But, luckily for us, gaming's most reliable ruler Mario Jumpman Mario is here for us to use. At 155 centimeters, Mario allows us to convert in-game measurements to real-life units, and, uh, shockingly, the courts in-mario tennis aces are surprisingly realistic.

I found the in-game courts to only be a few centimeters apart from the dimensions of regulation-sized tennis courts, which are usually, Usually 78 feet long, 23.77 meters, and for singles matches 27 feet wide, or 8.23 meters. I mean, the French Tennis Federation has already disallowed skin-tight body suits, so I would expect flying clown cars to also be deemed illegal in the game, but heck, at least the court sizes are in accordance with the rule books. Also, the game does us a favor by providing the speed of serves in miles per hour, which is similar to what's done in real-life televised tennis matches. However, a season theorist like myself knows to take these in-game space, speedometers with a grain assault. So how does the radar gun in Mario Tennessee's stack up? Using frame rates and more pixel measurements than I could shake a racket at, I found that the actual speed of any given serve was around 10%

slower than the speed that was boasted on the radar gun. And while that might seem like Team Nintendo is simply juicing the numbers in their characters favor to make things more fast-paced and exciting, the same thing actually happens in real-life tennis matches. The average speed of serves is always less than the speeds listed on television. And for one, It's not because we're intentionally being lied to. Instead, the radar reading is always taken right after the ball is hit. At that precise moment, the ball is traveling at its single fastest, before it steadily starts slowing down due to air resistance and spin. So, unbelievably, Nintendo is actually 2 for 2 when it comes to accuracy in this game. And we might as well make it 3 for 3 because I also checked gravity based on the speed of the falling tennis ball during serves,

and that checked out 2. 9.7 meters per second squared, which is pretty darn close to the real-life 9.8. Overall, this might be the single most realistic Mario game ever, provided you're willing to overlook the giant sentient ball of metal that's allowed to play tennis. Taking a closer look at how the tennis rackets break in the game, even before doing any calculations, one big concern stands out to me. If you zoom in during a successful zone shot, you can clearly see that the racket is breaking at the handle, the thick part where the player is holding the thing. This is an enormous,

red flag because in real life when rackets break they almost always occur at the strings, frame, or neck. Physically speaking, this makes the most sense because they're the weakest part of the racket and are thus the first to break when any sort of force is applied, either by the ball or when the player decides they need to rage quit. And oddly enough, in the image that eventually went viral for other reasons, Nintendo shows Luigi's racket breaking at the correct point, the strings and neck. But apparently it was just too hard to implement into the actual gameplay and so we're stuck with what's to picked it in the game, which I guarantee is gonna yield us some absurd numbers. Now, you might remember from the Blue Shell episode that impulse is one method of determining the force transferred by a projectile. That's the measurement here that's gonna help us determine whether a ball can break the racket. Impulse is equal to

force times time. It's also defined as the change in momentum, which for a ball traveling in one direction and then being hit in the other direction is gonna be pretty darn easy to find. Momentum is equal to speed times mass. So we need to find the mass of the ball, speed headed into the racket, speed headed the opposite way away from the racket, and the amount of time the ball is actually in contact with the strings, giving us a total of four unknown variables that are all pretty easy to find. We first have to assume that the ball has a regulation mass of a tennis ball at 58.5 grams, but considering how realistic the measurements already are in this game and the fact that the ball behaves similarly to a real ball on three different styles of courts, this isn't that hard of an assumption. The speed of the ball can be calculated using frame rates and pixel measurements just like we've been doing all episode, and we've been doing all episode, and we've

We get that it's traveling 96 miles per hour before impact, 43 meters per second, and 28 miles per hour, 12.7 meters per second, in the other direction after being hits. So with all those numbers in place, we're able to calculate the change in momentum. Final momentum minus original momentum. Because mass always has to be in kilograms for this, we shift the decimal point over a few places, which is always like the most obnoxious part of these sorts of physics equations, keeping your units together. I just gotta say that to express what I'm sure is the feeling of countless students taking high school physics. I would say more points and half points are lost on tests that way than any other way.

And speaking of those really minor things that are always super obnoxious to keep track of, this is the thing I always have to remind myself when it comes to momentum. The ball is headed in opposite directions before and after the impact, which means that one velocity has to be positive and the other one has to be negative. I mean, think about it this way. If they were both positive, that would just tell us that the momentum had slowed down, but it was still in the same direction, which obviously isn't the case. The total velocity change was the speed going into the racket, plus the addition of 12.7 meters per second added to it going in the opposite way. So in total, the whole thing changed speed by 55.7 meters per second. Does that all make sense? Welcome back to school, everybody.

Tell your math teacher about this episode, maybe you'll get to watch it in class. Anyway, doing the math gives us 3.26 Newton seconds, which is stupid and means absolutely nothing to any normal human being. So let's put it into terms that actually makes sense. To do that though, we're gonna need to know the impact time, which is as simple as counting the frames that the ball is in contact with the racket. It's only three frames, and at 60 frames per second that translates to 0.05 seconds, which again, shockingly is about how long a typical tennis ball in real life will be in contact with a racket strings. Mario Tennis Aces, ladies and gentlemen, I'm giving it the award now. Nintendo's most realistic game. Boom, there's the sticker. Slap that one on your box art.

Interestingly enough, this 0.05 number tells us that Mario and the crew are pretty much professional level tennis players. You see, players can control the impact time between the ball and the racket by adjusting the stiffness of their string bed. Beginners might not be able to hit the ball with a whole lot of strength, so they'll typically use a racket with a soft and low tension net to increase dwell time, the length of time that the ball is staying in contact with the strings. This in turn enables a sufficient impulse to bounce back the ball at high speed, but it comes with a price, a loss of control. In contrast, professional players are already striking the ball at full strength, so the They tend to prefer a racket with a tight net to shorten their overall impact time. Closer to the times that we see present in Mario Tennis Aces. For them, it's all about spin and trick shots.

In fact, many will actually keep two or more rackets on hand with different string bed tensions so they can actually change strategy mid-game. Anyway, with all that information, we can determine that the ball during a zone shot in Mario Tennis is delivering 65 newtons of force onto the racket, which is really small. It's only about the force of an average cat sitting in your life. So, how does that compare to the force required to break a tennis racket? Well, unsurprisingly, racket manufacturers don't really advertise how much force is required to break their product. But we can calculate a rough estimate of the force by looking at the racket's ultimate sheer strength,

which is basically a measure of the stress that a material can withstand before it snaps. Looking at the Mario racket in-game, it appears similar to the rackets used by professional tennis players, allowing us to narrow the possible options down to three common materials, aluminum, titanium, and carbon fiber. In the sheer strength equation, force is equal to the sheer strength of the material times the cross-sectional area. This is why rackets in real life tend to break at the neck or rim because of the smaller cross-sectional area But again, as mentioned earlier in the video, the racket in game breaks along the handle so the cross-section area is 7.4 cubic centimeters Adding to the difficulty of Mario's situation, most real tennis rackets are hollow. However, screenshots of the game as the tennis rackets snaps in half clearly show that Mario's rackets are completely solid

Therefore, the entire cross-section is used in the equations, whereas it would be much less IRL. Now all we need to do is plug in the ultimate sheer strengths of each suspected material to find the estimated force. Aluminum, with the ultimate sheer strength of 207 megapascals, would require 153,180 newtons of force to break. Titanium, with an ultimate sheer strength of 380 megapascals, would require 281,200 newtons to snap. And carbon fiber, with the lowest sheer strength of 95 megapascals, would require 70,300 newtons to snap. Remember that the zone shot provided a mere 65 newtons of force. Not 65,000, 65. Literally thousands of times less than would be required to actually break a racket.

No matter what this racket is made out of, it ain't gonna break after a single shot, regardless of how much power Waluigi is getting from that rose in his mouth. However, this of course begs the question, how fast then would be able to be. Would the balls have to be hit in order to actually get the results we see in the game? So I worked backwards using everything we talked about today and brace yourself. Even I as someone who deals with absurd numbers all the time on this show was shocked. In order to break the carbon fiber racket, the easiest of the rackets to break the ball in Mario Tennis would have to travel over a hundred and thirty four thousand miles per hour. That is 60,000 meters per second.

The aluminum racket would require the ball to travel almost three or three and three and three and three. 100,000 miles per hour 130,000 meters per second, and titanium rackets would need the ball to be hit at a whopping 500,000 miles per hour. That is 240,000 meters per second. What would happen if a tennis ball traveled that quickly? Well, first of all, there's no way that you're getting a ball to reach those sorts of speeds in the first place. The world record for fastest ball is held by Samuel Grawth of Australia who hit a serve of a hundred and sixty-three miles per hour back in 2012. Nothing short of a particle accelerator could make matter move that fast. Additionally, air molecules wouldn't have enough time to move out of the way of the ball,

which would cause a nuclear chain reaction as our yellow little fuzzball slammed atoms together in front of it. Forget losing a tennis match and boo-hoo, I have a broken racket. No! The Mushroom Kingdom would be overtaken by literal Mushroom Cloud. Subal would not only COO the player like we see in the game, but likely kill every single person in the process. You, your opponent, the entire, If you actually tried to return this thing, the fact that a tennis racket is a class three lever would mean that the force from the tennis ball would be four times greater as it surges up your arm. It's likely that it would disconnect your arm. Forget playing Mario tennis, you'd suddenly be playing happy wheels. Added to the list loyal theorists, we've found ourselves yet another Mario game in which everyone dies. Oh, and uh, about that sticker that I gave you Nintendo's most realistic game. Yeah, I'm just gonna take that thing. Because when it comes to,

to sports, Mario don't mess around. But hey, that's just a theory. A game theory. Thanks for watching.