Sick of falling to death trying to make those perilous, long jumps?
Are you riding a Yoshi? Then worry no more, my young friend!
Just tried the patented Yoshi sacrifice! Merely
press the (spin) jump button to send your cute friend, who saved you from being eaten by Koopas when you were a baby,
hurling into the abyss while you make it safely to your destination! That's Dr. Hourigan's
wonderful Yoshi sacrifice, available now at your favorite diamond nickel store
Dear Nintendo, Hi, it's me--
Austin! And today what I have on my ruin your childhood agenda is Mario 'jumpman'
Mario, amphibian murderer extraordinaire, and I'm gonna be focusing on something really specific; the Yoshi
sacrifice. This is a move that originates all the way back in--*god is gonna make me feel so old*
1990. Nearly 30 years with the introduction of Yoshi to the Nintendo franchise in Super Mario World.
Since then it's more or less become a trope so long lived that it's appeared in almost every game you could ride Yoshi in if
you want is a recurring staple in the dorkly bits shorts, has several Super Mario Maker levels dedicated to using this mechanic, and
even has its own song, and it's been my utter and complete
Obsession for the past few weeks and I have been focusing my laser vision science brain on this
problem because it is
deceptively complicated. It's a great example of the fun things you can do with basic Newtonian physics,
and since this problem has been keeping me up at night, losing sleep, writing on the walls like a mad person
in the middle of the night when there's plenty of paper lying around everywhere. This is the madness
I'm going to subject you to join me in the abyss of Newton's
kilograms, meters, and seconds per second.
Okay, so the basis, all right, let's review. This is a Newtonian physics problem no pressures, no relativity,
just basic masses,
accelerations, and speeds, which means we have three principal rules to keep in mind. The three laws of motion,
as laid out in Sir Isaac Newton's treatise philosophiæ naturalis principia Mathematica,
and they are as such: corpus omnipresent
Stop it! You literally did the exact same joke less than a year ago in the Super Smash Brothers episode. Bad Austin!
Alright, okay. You're right. Okay. real quickly,
the three laws of motion are: (1) An object either remains at rest or continues to move at a constant
velocity unless acted upon by an outside force. (2) The rate of change of momentum of a body is directly
proportional to the force applied and this change momentum takes place in the direction of the applied force. And
(3) When one thing applies a force on something else, the 'something else' exerts a force back on the first thing. These three ideas
are all we need to figure out if this Yoshi
sacrifice jump is
possible. And that's right. I did say
possible, because the first hurdle I thought of was, well, this is a closed system.
Is it even possible for Mario to jump and change his velocity if he's part of a closed system in the air? And in order
to answer this question, We need to do a thought experiment. Imagine instead that Yoshi and Mario were floating in orbit--
Okay, no way. Okay. They have spacesuits on *ok, all right, that's better.*
Anyway, if Mario pushed against Yoshi, would he be able to float away? And the answer is yes.
He would. The important thing is that not only would Mario float away,
He'd also send Yoshi floating off, because in order to move himself, he'd have to create an equal and opposite force
So Yoshi would go floating in the other direction.
You can even test this for yourself if you want. Just grab something for your table or whatever, and jump in the air and while
You're in the air, throw the thing that you grabbed up into the air.
I chose a pen when I did it, and there's two things worth noting. One:
It is surprisingly difficult to time your throw
correctly, and two: The pen, while it might not seem like it, is actually making me fall to the ground faster than I would have
if I hadn't thrown it, since it is exerting a downward force on my body in order to send itself up. This will become very
important later on. Okay. So the jump is possible, End of episode I guess! Hey, oh, wait, no, no, no,
because this maneuver is way more complicated than two objects floating in space or me making a of myself in my apartment.
There's lots of factors here,
So let's start at the most simple step first, The physics of Mario's jump, and then we'll build up from there.
Okay, so here's the earth and here's
Mario, and since Super Mario World is the first game that has the Yoshi sacrifice mechanic,
this is the game whenever possible that we're gonna be doing all our measurements in, so! First things first, we gotta jump. Good job
buddy! Using Mario's canon height,
1.55 meters, we can figure out exactly how high he's jumping each time in this game--
3.69 meters. It takes him 0.5166 seconds to reach peak height and it takes him 0.45
seconds to accelerate downward and hit the ground. With all the information, we can figure out the surface gravity of the game,
which is 36.419 meters per second squared, almost four times Earth's gravity.
Now the pennants among you are gonna point out that Mario does reach a terminal velocity where he doesn't accelerate anymore, and that all the
acceleration
downward is happening in the first few seconds as he's falling, to which I say, if we go by that metric, the surface gravity of
our world would be over two hundred and forty meters per second squared or over twenty three times the surface gravity of Earth. That is
over ten times the surface gravity of Jupiter,
so thirty six point four it is! But here's where the fun part starts, because knowing all of this but you figure out Mario's initial
velocity when he leaves the ground, which is important because after his feet leave the ground,
Mario can no longer impart any real energy into his body, and his velocity is locked in and the only
thing acting on him (aside from negligible air resistance) is the downward pull of gravity.
Thankfully, all we have to do is take this formula that equates potential and kinetic energy, and rearrange it so that our unknown
velocity is on one side. mass cancels out, and we get the square root of two times gravity, which we have, times height,
which we also have, all of which gives us a result of sixteen point three eight meters per second or
36 miles per hour. That is quite the jump, but we are still not done because all this work only gets us one piece of
a three-part puzzle
We need to figure out what we really want for reasons that will become clear later,
Which is the force Mario is exerting on the ground, and for that, we're gonna need force equals mass times acceleration
You see, when you're standing on the ground,
You're actually exerting a force on the ground.
But let's simplify this. A 1 kilogram mushroom sitting on the ground in the Mushroom
Kingdom is exerting a force on that ground, which is mass times acceleration due to gravity one times
36.4 is 36.4 newtons.
The ground is in turn per Newton's third law,
exerting an equal and opposite force on the mushroom, creating a net force of zero, letting the mushroom stay still. In order for it to
Jump, well, it would have to grow legs, but let's just imagine it can jump without them.
It's gonna have to exert a force on the ground that's greater than the pull of gravity. If it exerts just
38.4 Newtons the ground will push back with a force of
38.4 Newtons giving the mushroom a net upward force of two Newtons once you subtract gravity, which is enough to cancel out the pull of
gravity, and accelerate the mushroom up a whopping 1 meter per second per second not super impressive. But,
acceleration, force, these things need time in order to really mean anything to us. And in fact force equals M*A can be written
longhand as force equals mass times change of velocity over time in order to get any
meaningful data from Mario's jump and the amount of forces exerted on his body,
We need to know how much time it takes him to jump.
We need to know this for a few reasons; if we look back at that mushroom--
If you pop it into a vacuum and push on it with 2 Newtons for 1 second, It'll accelerate to 2 meters per second.
But if you apply that force continuously forever, it'll reach near light speeds in less than 150 million years.
*pause* Hi there, everyone future Austin here from my desk. While I'm editing this video,
I realized I made a mistake and I have actually made it in the past before, and instead of just patching it out,
I actually thought I would explain what happened really quickly before somebody says anything.
I mixed up my units because I just divided the speed of light by two
for the acceleration, and the problem is that gives you
149 million
seconds, because the acceleration is in meters per second
The reason I thought I'd be interesting to talk about this is because a lot of people mess up their units when they're starting to
do this kind of math,
and I wanted to show the errors that those can cause, and how really easy it is to do.
So if you actually divide by the number of seconds that are in a year,
which is 31 million five hundred and forty (thousand) seconds, you get four point seven five years,
which is a huge difference from 149 million years.
Anyway,
I wanted to correct that. On with the episode! *unpause* ...Million years! In short, in order to know how much force Mario is exerting in his
jumps, we need to know how long it takes him to reach the speed of sixteen point three eight meters per second, And while games have
Frame-perfect reaction times and he jumps instantly, this would create
incredibly high forces. A more realistic model, determined experimentally
by athletic scientists, suggests that vertical leaps impart force into the ground over the course of about point five seconds,
which means Mario's going from zero meters per second to sixteen point three eight in just 0.5
seconds, which gives us, finally, an
acceleration of thirty two point seven seven meters per second per second.
Now we just need Mario's mass which I figured out to be eighty nine kilograms already a while ago, using literal rocket science,
which gives Mario's jump force a whopping value of two thousand nine hundred and seventeen Newtons. MATH!
*Loud exhale*
Now that you know the basics, let's get back to Yoshi, because this is where things get really interesting.
We actually have to calculate two separate values because Mario can jump off of Yoshi at two different points;
He can jump off at peak jump height, which is the most feasible, but more terrifyingly,
He can jump just as high while falling at terminal velocity, and even more terrifying, I found out that the height difference,
that is the difference from his feet at the start of the jump to the top of the jump, is actually higher if he's jumping
off Yoshi. For whatever reason, Mario can jump over five and a half meters if he jumps off of Yoshi's back.
Oh, em, gee, this gives us an initial velocity of 20 point five meters per second
we have to reach in just point five seconds. Now, at the height of Yoshi's jump, this means he's exerting
3615 Newtons into poor Yoshi's back, and here is where things get out of control, because remember,
objects exerting forces on other objects create an equal and opposite force. The pen in my hand is pushing me down,
remember, and making me fall
faster. Imagine a nearly
200-pound plumber is jumping off my back in midair. That is going to send me
FLYING. And in this case, me is Yoshi. Yoshi would go flying downward, But by how much?
Well, that depends upon Yoshi's mass. Is Yoshi the pen, or is Mario the pen? And finding the answer to this was not
easy. Mario Kart puts Mario and Yoshi in the same weight class,
so they're probably pretty similar, but that's not a lot to go by. Finally,
I figured it out. If we can find an object that exerts uniform forces on both of them,
We can determine how much more one weighs than the other, if at all, based on how much they move.
Yes, that object exists, and it's the springboard.
Unfortunately, the springboard in Super Mario World does not affect Yoshi if he's off your back,
but it does affect Yoshi
independently in Super Mario Maker. Mario Maker isn't the exact same size as Mario World, but the physics are nearly
perfectly scaled pixel by pixel, with about 5 percent margin of error.
Not perfect, but it's what I got,
and guess what? They do pop up at different heights when left alone. Yoshi travels approximately
16% higher meaning he weighs sixteen percent less than Mario which means he's about seventy four point six kilograms,
which is bad news for this little guy, because he's already going to start accelerating
downwards at thirty six point four meters per second per second due to gravity, plus an extra forty eight meters per second from the thirty
six hundred and forty Newtons Mario's putting into him, meaning he's going to be hitting the ground at an Earth-shattering
Forty-two point six meters per second or over ninety-five
miles per hour. That is like a car crash.
But! It gets even worse, because Mario can pull the same move off while traveling downward at terminal velocity
Thirty point seven five meters per second, which means in order to jump up five point seven seven five meters
He has to impart enough force to cancel out all
89 kilograms of the Plumber traveling downward, and then enough to
accelerate him up still! If we take Mario's initial velocity of twenty point five and subtract the negative velocity thirty point seven five, his velocity
downward, we learned that his initial speed leaving Yoshi has to be
51 meters per second, or over
100 miles per hour, which would require an acceleration of over a hundred and two meters per second per second, which would require
9121 newtons of force, which would accelerate Yoshi
122 meters per second per second in the opposite direction, leading to a top net speed of over 91 meters per second,
which is, get this, over
205 miles per hour! That is like
accelerating a McLaren F1 to almost top speed, and then driving it straight into a concrete wall, and
that's completely neglecting the pressures exerted on the bones of Yoshi's back
that would likely completely powderize his spine from the jump alone
if the fall didn't kill him, and also, if Mario's capable of exerting 9,124 Newton's of force
Then it'd mean his top jump height from the ground should be over 36 meters, or a hundred and 18 feet!
Why aren't you jumping like this all the time?
So yeah, it turns out that yes, this move is theoretically possible,
It's just no matter what, terrible for Yoshi. Mario is such a jerk--
I mean whether it's being mean to his brother, sending his pets off to die in
bottomless pits, or sending them slamming into the ground faster than a speeding bullet so that they turn into dinosaur bone. Mario
Consistently is the most evil character Nintendo has ever made, and that's the kind of fun things you can do with classical physics.
Sincerely, Austin.
You
