The Perfect Goal Kicking Angle - Numberphile

Ben Sparks reveals the formula for optimal goal kicking... See brilliant.org/numberphile for Brilliant and 20% off their premium service & 30-day trial (episode sponsor)... More links & stuff in full description below ↓↓↓
This is based on the sport of Rugby Union. The principles could be applied elsewhere - why not try it?
Patrons can "enjoy" more footage of Ben and Brady kicking (and missing) rugby goals - / 80100608
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[@numberphile]

Patrons can "enjoy" more footage of Ben and Brady kicking (and missing) rugby goals - www.patreon.com/posts/80100608

[@Smilingcrittersall]

Hiii

[@andrewkepert923]

It’s OK - we only *wanted* to see the editor’s choice of kicks where Brady scored and Ben missed.

[@Vanta_Blue]

How to *really* sell your Patreon.

[@afratis2188]

I have written some mathematical papers and I was wondering if yall could do a video on one as I think it would be quite interesting for yall to take a look at if you want more info you can reply to me on this comment Psst! I think I could've created one of the largest numbers accepting the rules of the big number duel rules.

[@omarvelazquez1632]

"Maths is the art of reasoning from bad drawings" - Numberphile, you have a new T-shirt

[@DarklordZagarna]

So I guess, applying Erdos' famous axiom (a mathematician is a machine for turning coffee into theorems), that mathematics is the art of turning coffee into bad drawings.

[@OneTrueBadShoe]

I love Ben's presentation skills. He makes it easy to understand. That's why he's one of my favorites.

[@Henrix1998]

Surely someone has a database of kicks and it probably lines up quite well with this result

[@MrFrondoso]

Sports database are often premium or proprietary. Too bad.

[@jamescollier3]

lol. engineer vs physicist.

[@codycast]

Yo mamma has a database.

[@MrFrondoso]

@@Ms.Pronounced_Name nope. That's not what I'm saying.

[@peteradams2063]

​@Elan Cook There kind of is one already - Secret Base. They've got really interesting/funny videos on sports stats.

[@iansimcox]

The 30 degree launch angle is interesting information too. It means if you wanted to practice launch you put the ball on the 5m line and aim to hit the crossbar

[@RFC-3514]

Another factor is that, even for a professional player, shooting _harder_ (meaning the ball goes further) often implies a loss in _accuracy_ (on top of the reduction of the target's angular size). So, even if they could still clear the crossbar, they might lose so much control over direction and bend that they'd miss the posts. But, of course, to model that you'd need data from specific players.

[@reidflemingworldstoughestm1394]

@moi2833 All of them.

[@RFC-3514]

@moi2833 - Which part of the players do you want to measure, exactly?

[@1FISH]

@@RFC-3514 The part that shoots the hardest

[@johnchessant3012]

8:28 I love that he organized the presentation so that the listener delivers the key insight.

[@briandeschene8424]

I believe he is a teacher vs being a mathematician. So his focus is likely for the benefit of the recipient rather than the theory of the subject matter itself. Just a guess.

[@timseguine2]

As sure as the sun rises in the East, Ben Sparks will have brought a GeoGebra file.

[@MisterMajister]

Ben Sparks is such a legend. I hope we get many more videos like this from him!

[@Bodyknock]

Regarding the range of kicks, rugby fields are 70 meters wide, so the maximum the distance x could be from the center of the posts is 35 meters. That makes the maximum 45 degree hypotenuse you would kick is about 49.5 m. Professional rubgy players can kick over 50 meters in distance so basically if they're placing the ball right at the sideline and kicking at the 45 degree angle it's doable but probably almost at the edge of what they can do with any consistency.

[@Hiltok]

They typically take it back less than 35m when a try is scored right on the edge of the field. This suggests that there is an important trade off between apparent angle and loss of accuracy as kick distance is increased.

[@Jivvi]

Jason Taylor could do it. He was insanely consistent and regularly kicked conversions from the sidelines. He even had a season where he was 100% for the whole year! (1994 if memory serves) He was a rugby league player though; not sure if the width of the field is the same.

[@geckoman1011]

I have coached basketball, and it always blew my kids minds when I told them basketball was mostly just geometry. Everything from passing to shooting can be made easier (or harder) based on angles to the defense or the basket. Similar to the video, shots with arc, the basket looks bigger to the ball than flat shots.

[@rmsgrey]

Which is also why bouncing the ball off the backboard scores more often than a direct shot - the ball loses a chunk of horizontal velocity, so is travelling more vertically.

[@billr3053]

Not to put down the kids’ thinking, but why wouldn’t it be geometry? Everything is. Reality is physics and can be measured and understood / modelled with math.

[@SirWilliamKidney]

I love this, we need a giant match with the whole numberphile family!

[@liquidgeorge]

I can imagine Hannah being ultra competitive.

[@General12th]

I don't think Dr. Conway would help very much.

[@koenth2359]

​@@General12th No problem, put mr Sloane in the other team, to make it fair

[@koolguy728]

Another note on the "optimal" launch angle of 45 degrees that's often quoted. That value ignores air resistance, and it turns out that when you factor in air resistance the optimal range angle is in the 30s (almost in line with optimal leg power angle)

[@Jivvi]

That's interesting. Now I'm trying to remember what launch angle Mark Rober used with that kicking machine (where he wouldn't be limited by what's optimal for a human leg).

[@koolguy728]

@@Jivvi Oh yeah that's interesting, if you find it let me know

[@grempal]

Love this video. Reminds me of my best conversion ever. It was a very wet game with a couple inches of standing water on the touch line. Wing scored a try at the left touch line. The only dry spot to place the tee was between 5 and 7 meters from the try-line right on the left touch line. I kick with my right foot and have a natural ability to bend my kicks. This ability to bend my kicks let me aim at the far upright or just outside of it and the curvature of the kick increased the effective angle between the posts because the ball will be approaching the uprights more like a central kick. Anyway, I nailed the kick dead center between the uprights and got a compliment from the sir.

[@lordvenusianbroon]

The opposing team are also allowed to try and block the kick....so you also have to take account of how fast another player can get to between the ball and the posts, near the kicking spot, in the time it takes for the kicker to 'approach' the ball ;-)

[@davidsimpson1150]

Not in rugby - the opposition cannot interfere with the kick, unlike American Football

[@falkkiwiben]

@@davidsimpson1150 Yes, you're legally allowed to charge a conversion when the kicker moves forward. You're not allowed to do this when taking a penalty kick however.

[@davidsimpson1150]

@@falkkiwiben oops sorry, my mistake - you are right!

[@falkkiwiben]

@@davidsimpson1150 Np at all! Does not often come up so it's almost a secret law. I always enjoy teaching people the laws of this wonderful game :D

[@philw6056]

@@falkkiwiben That basically limits how close you want to be to the touch line. Whereas your physique and training limits how far you can go before the strength needed to clear the crossbar is at the expense of accuracy.

[@najakwarkle]

Such a fun episode. Nice work to both of you!

[@fatmn]

The fact that you showed a rare Justin Tucker missed kick during your American Football clip segment makes me VERY happy as a Steelers fan 😄

[@iseriver3982]

The fact that you can pretty much describe anything physical with maths is pretty cool. Even something as benign as walking out and kicking a ball.

[@nosy-cat]

"It gets messy, so don't do it unless you're brave" - oops, paused and tried (and cried) just before Ben said that.

[@noeatnosleep]

Ben is my favorite guest. Also I have to wonder if I'm the only one who feels like they have an aggressive instinctual understanding of this concept. I feel like I could probably get within a meter or two of optimal just by walking away from the try.

[@_SpamMe]

Never played rugby in my life. First instinct was "should 45°-ish, surely" ... quite pleased with that, in the end.

[@Jivvi]

I have played, but I wasn't a kicker. I thought it was going to be some sort of curve resembling a parabola, but intersecting the goalposts, so the apex of the curve is missing and it's undefined when you're in front.

[@Cowboymeisje]

Nice touch with the Hilbert ball. You can always make it one size bigger.

[@JustinoElArtista]

This is one of the beautiful things about sports. Athletes aren't necessarily doing equations in their heads, but they definitely are using their kinesthetic intelligence to understand and solve these angles. The best athletes aren't just the strongest or the fastest, but also mathematicians. Proper mathematicians, on the other hand, can give language to the rule of thumb athletes use so that those without great kinesthetic intelligence can understand what's going on.

[@jurgenpeters1373]

The explanation of the 45° rule was great. The actually kicking was hilarious bad.

[@dsacton]

Fantastic explanation!

[@andrebenites9919]

I have seen an article about it. It was talking about where would you look at a picture at a certain height that would give the maximum angular size. I remember I used the same logic in Handball to calculate the best place to score a goal on the sideline. I think it was around the 9m line

[@davidg5898]

I had one physics professor that gave us the same problem weekly during the term -- maximize the range in a specific direction of a projectile with certain givens. With each iteration, he introduced another real world factor to up the difficulty. By the end, we had an insane equation covering air resistance, wind, Coriolis effect, and some others. Long story short: If you're throwing spherical chickens in a vacuum, 45° is ideal. For pretty much all real world scenarios, expect it to be in the 25°-40° range. If you have a wicked backspin on your golf swing, you might even need to go as low as 15° (Magnus effect). So I'm not surprised that 30°-ish is considered ideal for launching a rugby ball.

[@linuslundquist3501]

I'm suddenly very curious about the setup for that problem...

[@davidg5898]

@@linuslundquist3501 The setup itself was simple. It started in a vacuum. Then added air resistance. I forget what order everything else came in. The terms just got tacked on since they can all be treated independently in the setup (when solving for a specific scenario, the trick is knowing which terms to combine or not combine). The final equation we had at the end of the course was probably similar to what military targeting computers use for long range artillery. 😆

[@linuslundquist3501]

@@davidg5898 Simple or not, I'm curious about the spherical chickens

[@davidg5898]

@@linuslundquist3501 You've never heard that before? It's from an old physics joke: A farmer's chickens stop laying eggs. He contacts the local university that sends out a veterinary professor. The vet examines the animals, takes blood and tissue samples, tests their food, and ultimately concludes that nothing is wrong with the chickens. So the university then sends out a bioengineering professor. The prof takes all kinds of measurements and observes the chickens walking around, tries out a couple of prosthetics and modifications to the chicken coop but with no change. Finally, the university sends out a physics professor. The physicist looks at the chickens for about a minute, then starts furiously writing in his notebook. About two hours later, he turns to the farmer and says, "I have a solution, but it only works for spherical chickens in a vacuum." It's funny because that's how physics is taught. Early on, everything is treated as a point mass or basic geometry (circle, square, sphere, or cube) because that's the easiest to model mathematically. In more advanced physics, you simplify problems by eliminating all terms except those absolutely necessary -- leading to a model that only sort of approximates the real world, but still gives a viable solution with an acceptable margin of error. Spherical chickens in a vacuum are important in physics.

[@jorgechavesfilho]

"Maths is the art of reasoning from bad drawings". So true!

[@ItachiUchiha-ns1il]

Such a well presented and informative video

[@elmerp]

My initial intuition was also 45 degrees, but could not really explain why. Thank you for backing it up with the maths!

[@archaeologistify]

I don't watch sports but this was so well presented I stayed watching till the end.

[@makotowang4363]

Me to

[@EebstertheGreat]

To answer the question about kicking in American football, you do get to kick extra points from the middle of the field (the middle of the 15 yard line in the NFL), but field goals don't work that way. When kicking a field goal, you snap the ball from wherever it was downed to someone who holds it on the field, and the kicker kicks it. However, if the ball is ever downed near the side of the field rather than the middle, it is moved toward the middle anyway to the nearest hash mark. So you never really have to kick from a significant angle. The most extreme possible angle (excluding oddities like dropkicks) would be from the hash mark at the goal line (or, say, half an inch back from the goal line), which is 18'6" from the centerline and 30' from the end line (where the uprights are), which is a 32° angle if you want to get it through the center of the uprights. But nobody actually kicks from the line of scrimmage; they snap it back 7-8 yards so that it doesn't get blocked. That makes the distance to the end line 51'-54' and the angle less than 20° even in this improbable worst case. So angles are not really a problem in the NFL. BTW, when you take air resistance into account, the optimal angle of elevation for a projectile to maximize distance is always less than 45° anyway. So that's not just about the biomechanics of your leg.

[@MrMas9]

Another banging video from Ben!

[@IlTrojo]

If you measure x from the closest goalpost to the ball rather than from the center, you can write theta as the difference of two arctans. Using the tan(a - b) formula you then get a formula for tan(theta). As long as you remember that tan is increasing, you might as well maximize tan(theta) rather that theta itself, and this cuts down the mess taking the derivative by a huge factor.

[@IlTrojo]

Also note that if you contruct a right triangle as: the hypotenuse is given by one vertex as the goalpost furthest from the ball, the other as the symmetrical point to the goalpost closest to the ball with respect to the ball line, the ball line is the hypotenuse's height; then the location of the right angle vertex is where you want to place the ball. This comes from the formula you find for the distance from the goal line, which coincides with the height lenght of the right triangle.

[@bradleysampson8230]

Great video, thanks Brady!

[@alandickey4787]

Another score by Brady and Ben! My fav team ❤️

[@JMUDoc]

It's an interesting exercise to do this the "long way" - I needed the tangent angle-difference formula, but I got the same result. Thankfully😁😁

[@jacobyarinsky4047]

I love how Ben always simplifies for the layman. He’s an engineer at heart. Btw… when did Brady become an official mathlete?

[@5eurosenelsuelo]

I literally worked on this problem while in university. What a kick from the past! I did it differently though. My idea was to assign scores to the angle and distance and the sum them up. Optimization is such a cool thing. Again, Ben's videos are just special.

[@DarklordZagarna]

ISWYDT

[@Geenimetsuri]

Fascinating stuff! Can't say I've ever even watched rugby for any length of time, but got me hooked :-D One minor nitpick tho, the 45 degrees is optimal only for a point-like object on the Moon! In real world you have drag, lift and you also have to consider the height difference from the center of mass to your target.

[@JavierSalcedoC]

This is Rugby World Cup year! Huge shoutout to Los Condores (Chilean team) which classified for the first time to this event. Save the date for England - Argentina (sept 9th), it's gonna be intense

[@theMosen]

What's so special about that game? I wish Los Condores the best of luck!

[@JavierSalcedoC]

@@theMosen Argentinians hate English to their guts because of the Falk-- *ehem* Islas Malvinas war. One of original Top Gear's episode illustrates this first hand

[@theMosen]

@@JavierSalcedoC Half the world hates the English, they didn't make a lot of friends with their empire, and they still don't with their pride of it. If you want to talk rivalry, England Vs Ireland are playing tomorrow

[@Syntax753]

!What a fun video :) Thanks for explaining Rugby to me too - definitely English rules apply for that!

[@alanredversangel]

Please pass this on to the WRU.

[@nicolecozens5023]

Or just Owen Farrell. That'll do to start.

[@Rubrickety]

Of course, if it’s Rugby Union, now you’re getting into set theory.

[@AT-sd2gh]

Excellent video.

[@simonf8370]

And Brady finds his other forte in life. Nice kicking and as always Ben delivers a fascinating insight into modelling 😍

[@daddymuggle]

The most obvious large assumption (to me), is that the ball travels in a straight line over the ground. Ben glances at this by mentioning wind, but there's also the curling effect due to the way the ball is struck. The model shown is a great first approximation, but players are always going to make seat of the pants adjustments for wind and handedness, etc.

[@bobrong9645]

Great video, as often. And Ben, you should consider a coaching career, I mean, you would bring a lot to England.

[@daninaidu]

To be fair, I think anyone would bring a lot to England right now!

[@nottmjas]

Get him on the next plane to Dublin straight away

[@davidmilhouscarter8198]

0:38 In American Football, the scoring team can also opt for a two-point conversion. They get one attempt to advance the ball into the end zone from the two yard line. (About 1.83 meters)

[@RUBBER_BULLET]

The points in rugby union are all primes: penalty/ drop goal - 3pts; try - 5pts; conversion - 2pts; penalty try - 7pts.

[@sirgermaine]

My intuition was to imagine the goal posts are zero distance apart and that would put your best distance at the distance equal to your horizontal offset, then considering what effect widening the post opening has. This explanation is very nice in a totally different direction

[@jursamaj]

I'm not following your argument. How did you get from posts 0 distance apart to 45°?

[@Jivvi]

The only difference widening the goalposts has is that _x_ becomes smaller compared to _d._ But so does your distance from the try line. You still end up with an asymptote at 45°, and a curve that only becomes visible when you're very close to the posts. If _d_ is 0, you basically end up with everything being undefined. Two of your points for making the circle are the same point, and the angle is 0° no matter where you kick from. None of it makes sense unless _d_ > 0. It's like trying to kick the ball between a post and itself.

[@Jivvi]

Fun fact: a try in rugby was originally called a touchdown, and "try" referred to the subsequent attempt at goal. Originally the scoring was very different, and most of points came from the kick. This changed over the years, and when the touchdown eventually became more valuable than the kick, it started being called a try. American football never changed those terms (an extra point attempt is still officially called a try), but they did remove the requirement to actually touch the ball down.

[@chadjones1266]

Excellent thank you

[@nicolecozens5023]

I feel Brady is trolling the English by releasing a rugby themed numberphile video just after the England France game...

[@ulfy01]

"it's close enough for numberphile" Are you implying we have low standards here? 😁 In seriousness, that was a fun episode! I always enjoy seeing Ben.

[@zamiyaFlow]

They should show these on trigonometry classes, specially when students go "jeez when am I gonna use this in real life"

[@KuroroSama42]

The first thought I had as the answer to the question, about 2-3 minutes into the video, was "It's an isosceles triangle, right?" Turns out I was slightly wrong, but got the rule of thumb. The slight curve of the circle behind the goal posts means it's not quite isosceles, but it's a very small distance so it's almost isosceles.

[@millylitre]

The optimum distance is the geometric mean of the two distances from the goalposts to the try position, paced out along the width of the pitch at the goal line. Example, 10 paces from try posn to near post and 15 paces to reach the far post. 10 x 15 = 150. Square root of 150 = 12 and a bit. Walk 12 and a bit paces from the goal line to reach the optimum position.

[@Hiltok]

Using Ben's parameterization with x being the distance from the centre of field and y being the distance from the goal line, the apparent angle, theta, between the posts from the position chosen by the kicker is simply given by: theta = arctan((x+2.8/y) - arctan((x-2.8)/y) Taking the derivative with respect to y gives (after a bit of simplification): d theta / dy = -(x+2.8)/(y^2+(x+2.8)^2) - -(x-2.8)/(y^2+(x-2.8)^2) Setting d theta / dy = 0 for the maximum (and thus dispensing with the common denominator which is strictly greater than zero) gives: d theta / dy = 0 = 5.6x^2 - 5.6y^2 - 2(2.8^3) = 5.6x^2 - 5.6y^2 - 43.904 which is the hyperbola equation for maximizing the apparent angle. Of course, this can be simplified to: 0 = x^2 - y^2 - 2.8^2 = x^2 - y^2 - 7.84

[@ErinJayEldridge]

This is a fine bit of maths, but something else to remember: the angle you're trying to maximize corresponds directly to the apparent distance between the posts, which is directly observable (if maybe difficult to measure, but you can put a hand out at arm's length or something to get a rough measurement and that's sufficient to find a maxima). It may well be easier to measure that angle than it is to measure the circle radius x used in the maths.

[@millylitre]

Indeed so. Hold your thumb at arms length and estimate the width of the goal in "thumbs". When the number of thumbs levels off you are close to the optimum position. An eminently practical rule of thumb.

[@floyo]

You can rewrite the formula using x^2 - (d/2)^2 = (x-d/2)(x+d/2). Here x - d/2 is the distance from the ball to the first post and x + d/2 is the distance from the ball to the second post. Multiply these and take the square root and you're done. I think this is easier to remember

[@naverilllang]

Thanks, I'll remember this next time I need to kick a rugby ball

[@erbiumcanne3095]

Best videos with this maths teacher

[@mikescully6546]

Brady putting on an absolute kicking clinic!

[@jasonpatterson9821]

In addition to biomechanics, air resistance matters substantially for a real ball in the air, and an angle under 45° winds up traveling farther.

[@geoffstrickler]

Crossbar height doesn’t alter the analysis of “best angle”, but it does affect range. This is why kicking from inside the 5m line is generally a bad idea. So, if you’re not a strong kicker, you might need to choose an angle that gets you within your range. Wind is of course a factor as well, and it too might lead you to choose a shorter path with a slightly narrower angle between the posts.

[@Veptis]

thanks Brady for providing the B roll here!

[@carnage9029]

Watched this guy give a lecture at Greenwich Uni he's a great presenter

[@neuralbrew2976]

The best solution is don't overthink it. There is a strategy in American football of the opposing team calling a timeout just as the field goal kicker is about to make an important kick. The belief is that the more time a kicker has to think about the the kick, the more likely he will overthink it and miss.

[@Verlisify]

Brady has some sleeper talent. First time kick and let's not forget first time sinking the ball in the elliptical pool table

[@alistairkentucky-david9344]

It's called a "try" because you get a 'try' at goal

[@Thats_Mr_Random_Person_to_you]

About to say this. In the original rules, the try was worth no points. It only gave the the opportunity to kick for the posts which is what got you the points (and just 1 point if not mistaken). So yh, as you say the 'try' allowed to to try and kick for the actual points.

[@Jivvi]

@@Thats_Mr_Random_Person_to_you in the original rules it was called a touchdown and was worth 0 as you said. The try (the attempt at goal) was worth 5. Over the years, the value of the try was reduced and the value of the touchdown was increased multiple times, to where they were eventually worth 2 and 4 respectively. After rugby league split away, rugby went one step further and increased it again to 5, but rugby league didn't. American football went even further and increased a touchdown to 6 and reduced a try to 1, and still uses the original terms (an extra point attempt is still officially called a try), but rugby started calling a touchdown a try around the time it became more valuable than the kick.

[@ElvesofZion]

Silly correction, in American Football, despite the name , for a Touchdown you don't actually have to put the ball on the ground. It counts the moment you cross the goal (or try) line with an portion of the ball.

[@numberphile]

I filmed a whole section explaining that and, to be honest, it was starting to feel like unnecessary detail… wanted to get to the point of the video. :) I’m a keen follower of both sports. In rugby of course you do need to ground the ball.

[@ElvesofZion]

@@numberphile Certainly didn't need to be in the video, just a fun fact with the different kinds of Football and the evolution of naming vs rules

[@arrgghh1555]

It amuses me the one you don't need to touch down is the one called a touch down.

[@mralistair737]

@@arrgghh1555 you mean in the sport called football that doesn't use the feet or a ball?

[@88porpoise]

​@@arrgghh1555a long time it was required to touch the ball to the ground. The rules changed but the name stayed.

[@Piffsnow]

The angle we're interrested in is atan((x+d/2)/y)-atan((x-d/2)/y) The derivative with respect to y is (x-d/2)/(y²+(x-d/2)²) - (x+d/2)/(y²+(x+d/2)²) If we solve the derivative = 0, we get y² = x² - (d/2)² :)

[@Saltyarticles]

Coincidentally, this type of maths is used in calculating lateral soil pressure surcharges for soil retention systems. Amazing video.

[@yeeunpark8291]

Yo I love your videos

[@solidsn4ck]

Rugby and Numberphile, perfect!😊

[@Kaanin]

I'm a simple man, I see Ben Sparks, I click.

[@Simon-ph1nf]

Next international rugby game your going to see a bunch of team managers with IPads shouting co-ordinates to the kicker to get the best spot :) great work

[@petrospaulos7736]

I love how Brady is flexing!!!

[@oafkad]

They should use this rule in American Football. That's such a cool mechanic.

[@JohnSmith-kt9ys]

It is the rule. The hashmarks are 1/3 the field width in high school. Where you score is where you can take the snap. In college and pros the hashmarks get narrower. You don't get to choose where in the hashmarks you start the play. In high school the extra point can be at a pretty narrow angle if you score on the wing. Pro American football is not as interesting of a game as the lower levels oddly enough.

[@TomLeg]

So if you score your 'try' between the goal posts, your kicking position is the sqrt of a negative number, so you briefly disappear into an imaginary plane

[@bencollier9423]

Just kick vertically up ...

[@zlatanibrahimovic8329]

why do you need quotes around try

[@hyfy-tr2jy]

Gee...another Maximus the Mathematician video so soon!? We are not worthy...but we ARE entertained!

[@42ArthurDent42]

i won't spoil the formula, but when i was a rugby player, i used to just do x=y to decide my spot, not so bad

[@anschelsc]

Brady is so proud of his kicking skills

[@oneMeVz]

My intuition told me it would be 45°, and this just proved it. I think next would be to ask what's the optimal target between the posts. Like does the ball want to pass with equal distance between the poles and crossbar, or can you aim lower?

[@wyattstevens8574]

In American football, the crossbar is 10 feet above the ground with the height of the goalposts another 20'. The width, though, is weird: college and professional (think Super Bowl) goals are 18'6" wide, but in high school they're an extra 2'4" on either side. The kick is usually from between the holder's position at 12 (NFL) or 13 (NCAA/high-school) yards back to within 20. For clarification, (seriously), watch the SNL "Washington's Dream" skit! He shows how little sense the US customary system makes- and I think he's right! (For any SI users, 1 foot= 30.48 cm exactly)

[@JavSusLar]

Isoptic curve of the posts is an arc which must be tangent to the straight line. Bisect the segments between the posts, that line contains the center of the circumference. The radius of the circle is the distance from the try point to the centre of the posts.

[@JavSusLar]

The geometric locus of the optimal points is a hyperbola.

[@StephenGillie]

Oh, THIS is GeoGebra. It's available through WinGet - I've seen it go into the manifests, always curious what it looks like. Thanks for using it to show it off.

[@elliotwilliams7523]

Got the answer before I watched. I used calculus but got the same answer. Great provlem

[@mathewmunro3770]

The width of the ball & the posts makes the optimal solution slightly further from the goalposts.

[@ubertoaster99]

0:14 definite knock on!

[@matsnordstrom8584]

At the point of infinity, the angle of scoring under the posts or scoring at the corner is equal.

[@petejandrell4512]

On the thumbnail HILBERT on the ball instead of GILBERT, very drôle

[@inexp1]

wow. Just wow.

[@ivancota9762]

"Math is the art of reasoning from bad sketches" 😂

[@RibusPQR]

I never knew Brady is a good athlete. Now I want to see him up against Tony in a shootout! Also this field footage made me think of Taskmaster. I think Ben could do pretty well in that kind of show, although he'd have to leave the Geogebra at home.

[@harryharperable]

10:55 Neat! The optimum spot turns out to be the geometric mean of the distance from the try spot to the near goalpost and the distance to the far goalpost. 14:01 the rule of thumb proposed can be interpreted as the arithmetic mean of these two distances. Which is always a bit further than the optimum by AMGM inequality .

[@nbianco19]

This can be modeled with added shot accuracy compared to perceived width of the goal, and distance accuracy, all multiplied by the intrinsic accuracy of the shooter himself. /all modelled in 3d transparent models, in geogebra.

[@halcyonOH]

pretty cool

[@SchumiUCD]

It's interesting that this would indicate that conversions from the touch line should be taken about 35 metres out but kickers never go back that far. They always take those kicks from the 22 metre line. I assume it's because the distance (about 50 metres) is too long.

[@dominiquelaurain6427]

It is a classical problem ... in the vertical plane (viewing a tower or a painting) rather than horizontal plane (rugby rectangle field). If O is the corner of the field, because H1 and H2 feet of goal are very near then power of O with respect to circle is OH1 OH2 near OH1^2 and equal to OM^2 where M is the shooting point so roughly OM = OH1 and because field corner is rectangular, we get the 45 degrees angle.