Say a basketball is one foot in diameter. The basket is two feet inside diameter and is ten feet above the floor. The foul line is 10 feet in a horizontal direction (sqrt of 200 diagonally) from the center of the rim.

Only swishes count. No rim or backboard.

There is no air friction.

Your shot is perfect as far as side to side is concerned. It is only your speed that you worry about.

You can take the shot starting at any height as far as where the ball leaves you hand (ten feet away horizontally from the basket's center).

For any height you choose, there is an optimum upward angle to start the ball off as far as margin of error is concerned. There is a trade off. At a low angle the ball travels a shorter distance but the elliptical target is skinnier.

Assuming you use the optimum angle, what is the height where the percentage margin of error on your speed is greatest.

(you may ignore the lengthy discussion intended for me and some other people to get familiar with the problem, thinking out loud so to speak, and go to the blue part if you find it to be making sense right away, then return to the lengthy text only to decide on some details of realism and practical value of such exercise. Some others however may prefer the gradual discussion until they build intuition . Often setting the problem properly mathematically is the most important detail to save time even if the remaining hardcore math part still proves usually a tougher process.)


For future reference lets call A the point projection on the court plane of the point S the shot ideally must be made from (in some height h) in order to be at the 10ft distance from the center of the rim that we call C. So AC needs to be 10.

You also do not want the ball to hit the rim at all even if many of those hits are super safe and end inside (not wanting this makes it a different problem than the real basketball problem though, granted the real basketball problem is even tougher to gauge the exact range of rim hits that end inside still, so i understand the elimination but just saying we will miss something very important here in terms of realism unless we make it a prop bet clean shot).


Now the speed is a 3D vector (say pointing within some cone type thing - nor really but until we determine it lets call it that). So you want the height H for which that vector has the widest margin of error allowed (error in magnitude angles etc).

Now the problem with this is that realistically some errors are easier to do than others. Also one may (not sure without solving of course exactly) find a height for which the range of possible vectors is maximized (range maximization defined as say the maximization of the volume of the geometric locus/set of points defined by the end of the velocity vector as drawn from the point of shot at height H for which you have a successful shot eventually). But then is that really what you want? I mean yes it is what you want as error range but as i said some errors are easier to do than others. For example lets say that one finds that for some height there is a wide margin of errors in one angle and speed magnitude but not the other angle (we have 2 angles for a given trajectory even if you want to fix one as 0, its never going to be realistically true) compared now with another point that has smaller ranges but the errors are more similar (ie i mean say in one height the speed has 1% room for error, angle1 1%, angle2 1% and in another height has 2% for speed 2% for angle1 but just 0.3% for angle 2 making the second height - to be determined of course what the proper probability volume metric is - better than the first but from a realistic point of view its easier to have the same room of error for all 3 parameters instead of big room in 2 but tiny room for error in one making it essentially tougher even if still wider overall technically - unless we define it very well in terms of total probability and we do not care which parameter fails more often at all as long as the total probability is maximized).

So its a bit of a problem to leave things unspecified because of this issue. We could eliminate that by specifying the standard deviation from a needed ideal vector for each of the parameters that a human can deliver casually without extreme stress (speed magnitude and the 2 angles) ie say some 1%-3-5%? (to be decided later) and then try to maximize the probability to land a vector in the range needed ie with the tip of the vector in the safe locus of initial starting vectors, based on these distributions (say normal). So for every speed magnitude you will have some angle1 (or 2 ) that produces ideal shot. But then these will be coming with errors (distributions) hence a probability to land the final speed vector in the safe locus.

Another point is that also the height of the release is subject to error itself. So yes it may be ideal for some height but that height also comes with some error itself in real life unless this is not a shooter but a machine shooting. So in fact we need to have errors for 4 parameters.
1) Height
2) Speed magnitude
3) angle 1 (the usual angle the vector of speed makes with the horizontal plane)
4) angle 2 (defined as the angle the trajectory projection on the court line makes with the line CA joining the center of the rim C and the center of mass projection on court of the shooter A.

(or you could specify the 3 components of the speed with errors instead plus the height error) (in fact a 5th source of error is how accurately we place ourselves at the exact 10 ft from center of rim point that is parallel to the sides ie how accurately A is placed to the ideal point)


If we can agree on the formulation an attempt can be made to find the maximization of the volume or the corresponding probability to land in that velocity vector tip safe volume using the errors/sds of these 4 parameters placement.

Otherwise you see its not realistic to claim a human can deliver angle2=0 exactly and height = H exactly without any errors but have errors in magnitude and angle 1 (its not realistically good/self consistent even if simpler).

However if for some reason you insist on having errors only in angle 1 and magnitude of speed and have all else perfectly fixed as if by a robot that only moves the shooting angle and the speed magnitude, we still need to know what are the error sds (distributions) in these 2 parameters.

So either specify a human shooter with 5 parameters of placement and their errors , height, point A coordinates , v magnitude, angle 1 and angle 2

or

a robot with perfectly known H,A and angle2=0 with only 2 parameters left v and angle 1.



(you probably will aim for the robot option but then we have a perfect machine shooting and if we have a machine the problem is trivialized, but having a human with say a natural source of error in all parameters of the type say of some decided 1-2-3% is more realistic)

Solving such problems ought to help us learn something so what exactly do we want to learn here, something for robots or for people? (people cant obviously control perfectly A coordinates, Height and v,angle1,angle2) (they may however have a very small error in A as they stand initially perfectly placed and a larger but similar error in the other 4 variables)



So basically the problem after we have decided on the distributions that the human or robot produces with the parameters v, angles and height etc is the determination of the locus for the tip of the velocity vector for every height H that lands inside safely and then from this the calculation of the total probability (based on the distributions now) to deliver in practice a vector in that range. Then we maximize that probability for H.

By the way we know that as already you noticed, there will be some angle that minimizes the magnitude of the speed (hence smaller detrimental end effect of a fixed % error in magnitude) needed to get there. But maybe going a but higher allows the ball to come down a bit closer to vertical enjoying better effective cross section of safety. In doing that of course the effect of velocity magnitude and angle error is enhanced.

I need to catch some sleep but i may return to it during the weekend.

Lets first determine this;
what pairs v,theta1 (in the simplest of cases we will go for realism if needed later once we get some early intuition) allow a safe passage.

To do that basically describe the trajectory parabola of the ball center and the corresponding sphere around that point that needs to not be touching the rim at all times.

So for every x,y,z as functions of t and v,theta1,theta2, H etc we have a sphere that is running and which will have a surface equation that we need to not be intersecting the rim. Essentially we need a function that measures the distance d(t,v,theta1,H etc) between sphere and rim at any point in time and maintain that it is >=0. At the same time we need the ball to clear the basket which implies it needs to be crossing the interior of the rim. So we have 2 conditions, interior crossing (ie point of intersection of trajectory with the basket disk having distance from the center <R and then d>0. This will force the condition v,t, theta1 etc must meet at the beginning. This is also like demanding the distance of the center of the ball (x,y,z functions of t etc) and the rim is always >r (r the ball radius). So for every trajectory x,y,z the minimum distance of the center with the set of points of the rim must be smaller than r. We need the function that gives us this minimum at all times for a general trajectory. So the general geometric problem now is to specify what is the minimum distance of a random point x,y,z in space with a specific circle that has equation x'^2+y'^2=R^2, z'=L (1) (we have taken as origin (0,0,0) of the coordinate system the projection of the center of the rim C on the court floor, L is the height of the rim from the ground. We need to determine the min d(x,y,z;x',y',z') over that set (x',y',z') that is satisfying (1) and demand its always >r , r the radius of the ball).

Of course the trivial idea immediately one should have is that if you were to possibly do the unthinkable and unrealistic for humans ie go up a few meters higher than the rim level you would only need a smaller overall speed to send it like a waterfall (horizontal projectile) and the error in speed then would be likely more comfortable given that you have to produce a smaller magnitude (i assume its easier to control and make a small relative error when the effort is not very severe ie when the speed you give it is not huge, ie more of a finesse easier less energy intensive effort) and it falls also with a near vertical angle (certainly closer to 90 than 60deg).

Now of course as a human project this is not doable which likely suggests the problem requires a simulation numerical effort to solve within constraints of Height that match typical human skills especially given the fact that the error one has is a function of their jump position uncertainty at the moment of release meaning that you want to go high of course and be able to get a decent entry angle for a smaller overall velocity but the higher you try to jump the more the uncertainty in your release velocity probably (ie the error in speed is now function of the other details like how high H is ie compare a foul shot with a jump shot).

The more one thinks about it the more of a simulation numerical project it feels like especially if we model properly the error distribution for humans in V, theta, H etc which will further complicate things. (in the absence of such modeling it would probably seem like the higher one could go without compromising accuracy ie taller player or higher release hence smaller speed requirement to get a big decent entry angle the better it would get before doing any calcs). (smaller speed and great angle at entry is also an advantage for shots that hit the rim a bit as they have higher chance to go in instead of strongly rebounding far out, although this is not a factor under OP conditions here)

Compare also with the typical office waste basket paper ball games. Dont you think its easier to send the paper made ball in the waste basket from a high standing position than if you kneel down and send it up first?

How tall are you? Edit: At what height do you release the ball?

Yay for reading comprehension. Leaving everything aside about the actual body mechanics of jumping like a freak affecting stability (and that real basketball players don't actually optimize angle) and that with actual people error in angle is probably different in magnitude than magnitude of error in velocity, the answer is "at least as high as the hoop." (in real life, the granny shot is best, but that is because real life basketball players don't optimize angle)

Masque, your long posts were basically a question of whether we should be calculating absolute Vmin-Vmax versus whether we should be maximizing Vmin-Vmax divided by whatever V a perfect shot entails, correct (and the same problem on angle)? That, and some guesswork on the limits of human exertion that are pretty awesome in terms of practical physics.

I think what we are solving for here isn't solvable without a bit more information from DS.

I'd just release the ball from 20 feet above the rim, aimed just inside the back, traveling at 100 million miles an hour. Any faster speed is a swish and I can release at the same angle missing like 99.9999% of initial velocity and still swish. EZ game.

Of course TomCowley is correct. If we make allowable velocity a function of height, then there is something more to be said. But otherwise, below the rim one has a capped margin of error, with trajectories of the form:



And above the rim, say from 50 ft, it becomes clear that margin of error can be made arbitrarily large...



...where as , both and .

Yes i agree with Subfallen on simulations i also did on regular shots below the rim. I didnt explore the fast shot ones because they are unrealistic. So from above the rim i only considered small velocity horizontal shots that are very easy to produce.

Yes very fast you can use the analogy of a shot gun or laser etc. You just point and shoot and instead of bullets or photons (hundreds/thousands of times faster) it fires balls. But in real life how do you control the angle error if you have to produce the fastest possible human shot speed. Yes the speed magnitude has a big room of error but what about the angle now? If on high speeds the error angle (error function of speed magnitude due to intrinsic human aiming issues thn) becomes like 10-20% it fails even for this unrealistic way of doing it.

So far with simulations it seems the higher you get (without jumping to introduce other errors) and the bigger angle over 45 deg the better it gets (that by the way are not as realistic - but good enough for a clue- as the problem deserves if done truly scientifically because its not as simple as a crude plotting makes it to be that proves accurate enough for visual purposes but may fail to capture the real geometry of the trajectory at the neighborhood of the rim when in 3D more realistic error sense studied.)

In real life the human player has issues like how the angle is estimated when not able to aim from above. The entire problem is not well defined unless we study a bit better how human brain decides on a velocity vector and what the distribution producing that result happens to be. So you have the fluctuation around what the ideal trajectory is and also what exactly do one means with "ideal trajectory". How close in guessing the ideal speed is human brain and what is the distribution of error around that guess. Its like 2 distributions. Kind of like error in estimation and error in delivery of the estimate.

masque de Z -

Would it be fair to say this sort of problem is either: (1) a straightforward, although possibly tedious, exercise in calculus; or (2) a task for simulation, with no hope of exact solution?

(I mean, depending on how realistic one wants to be. It is hard for me to think of how to add realistic-type constraints without immediately putting the problem out of analytical reach. Although I am not adept with computer algebra systems.)

Edit - have a good night, I've enjoyed your posts in this thread!

Oh yes if we start trying to make it realistic enough we will have to introduce vast research efforts here no doubt because we introduce complexity without controlling its own range of theoretical "errors" and theoretical guesswork on what skill/delivery is etc. But ultimately it is what has to be done to be proper.

Otherwise yes it becomes a "simple" (yet tedious-lengthy) problem with V, Theta, H as only variables just the usual parabolic equation and no particular care near the rim to study correction in the visual trajectory sense (ie that would have you simply say land between x1 and x2 type thing with these x calculated only using radii differences type arguments instead of more complex math that takes care of the fact the orbit is not a straight line, it is curved and therefore you need to be thorough not only at the level of the rim vs cm of ball but throughout the path when doing it in 3D. I mean to some it may feel obvious that all you do is compare CM positions with rim edges and test if larger than r etc but you need to actually prove that is sufficient and you do not exactly need to study what is going on a bit earlier or a bit lower than the rim level y=L etc because i mean you have a curve not a straight line and the ball is not a point.

Actually guys you should challenge your friends (some prop bet for fun?) if you find yourselves in some home basketball court that has say a window nearby or some elevated place (ladder thing whatever) you can go and throw a ball from much higher than the rim (or maybe youtube has it already lol) and see if you can deliver a shot better by hitting fast as if aiming gun style or you still need to apply a horizontal projectile trajectory again for better more accurate placement. (be safe doing it though lol)

Or you can try the usual desk waste basket (or bucket) ball made of paper deal and see if hitting it fast aiming is better than the gentle throw option.

I am not so sure you will find that the fast strike aiming shot style way is more efficient (due to the violent nature of the motion to produce high speed introducing significant angle errors) unless you are like almost above the rim (and not very far) in some small angle that it helps your aim better this way (i mean the effective acceleration corridor, defining a better aim, your hand-s make).

And of course in general you want the x component of the velocity to be small so that the effective cross section of the rim (near ellipse but not exactly in general due to curvature of orbit ) is improved which enables for larger error margins in both velocity magnitude and angles.

Also the air resistance that we assume negligible dont be so sure its exactly so negligible for such fine tuned situations.

Real life points to a low starting point to get gentle bounces if you hit the rim.

The granny shot thing? That just shows that basketball players don't use the correct angle (they shoot at too low of an angle). Also, that backspin helps, and that one-handed shots lack side-to-side precision, and that using a pendulum-like underhand throwing motion allows more control over velocity and vertical release angle.

The answer is going to be that the best release point is at least the same height as the basket hoop (if both angle and velocity errors are allowed for). It is the same exact question as "what height basketball hoop would be easiest to swish from ten feet away."

Yes you too nice charts and visual examples. Do you have a physics simulation package or did you write your own programs? I did mine without any printout just to get an idea but yours is great because it can be shared by others nicely. Do you agree also that on large speeds the angle error becomes in real life a big issue when far above the rim because its hard to be so precise in angle when the speed/acceleration effort needs to be so explosive?

Yes i didnt directly respond to this because it was kind of made by the other posts anyway.

As for granny shots for guys like me that were never kids in this country to have heard the term lol;

this is what it is from some apparently local team (that makes it hard to be a fan lol) old timers hehe;



I bet this contrary to our results so far that you need to try from higher heights as much as possible and larger angles to avoid having very big velocities, especially the x component, and to enjoy wider entry cross section, this style is favored because it allows him better chance to control the precision of the velocity vector vs an upper handed shot and for the back spin reasons mentioned in case of failure to have a clean entry. Very few of course use it today. In fact i probably never recall it watching basketball in my life as something that would happen even 1% of the time?

And this is why real life sports physics is a true science of its own because its a lot more complex than any simple approach one may try initially out of fear of massive complexity and lack of proper modeling/data.

My own basketball skill is ridiculous by the way, i gave up practice after high school anyway and picked it up only a bit with department friends while at Stanford but nothing major only in parties. Well ok i can play a bit but nothing great. My mom used to take me at her school she was teaching physics (yes family connection with physics) when i was like 6-7-8 and i was playing baskeball alone out in the court during her classes and you can imagine how hard it felt for a little kid. I even sometimes have dreams of the ball being too heavy when i play basketball lol so bring on the analysis, add to that the dreams about never having shown up on a history final exam repeatedly - never the case- back in high school, effectively making fake/invalid all my post high school degrees earned (due to not valid high school diploma which was always requirement for any next level lol). I was actually the very opposite, record high grade point average student in real life and never had any stress in exams. By the way i was decent at History, i mean A to A+ type student even if i hated memorization of bs details, always preferring the real logic/meaning/connection of events rather than the dates crap and certainly overall not enjoying the history class (until the very late age 17,18 that it became a more thinking/culture/politics style class) as much as other topics.

Anyway regarding basketball you dont want me on your team, nobody ever did anyway even if i was helping them in the homeworks and never snubbing them in classes. I always though stood up to bullies and had no problem getting down and dirty with sobs that showed violent aggression outside class where they thought their world started and mine ended only to find out how not exactly right they were (but not very wrong either in sports lol). I was taking my revenge though on long jump, 100m sprints and even long distance runs (counter-intuitive as this may look due to the other ones). I sucked at classic (eg uneven bars) gym exercises though.

Uneven bars is a girl's event.

It is a girls' event in competitions, at school it goes both ways at least where i came from. By all means go ahead and tell me at 14-15-16 that its a girls' sport when you have a hard time at it together with 75% of the mixed sexes class of some 30 students lol.

Just about the only cool thing/bright moment at indoors gym class was the sexy looking bad girls that somehow had a tendency to "accidentally" touch you with their curves in tight exercise clothes while waiting in line etc lol. They even did that at a finesse higher level while getting homework assistance in math/mostly geometry during breaks etc at the complete disapproval of the "good" girls that were very easy competition in class but still wouldnt dare take themselves at such low level with me... asking help. Of course i was all too happy to help both lol. I even won the approval of the "bad" male students because the tough/strict math teacher had a hard time with me on occasion due to unorthodox complex proofs he was having a harder time following (which for the students suffering at his hands in blackboard examinations and grades, was their revenge against him, hence i was a folk class hero). High school, what a cool time for almost everyone!

We separated the girls out for that, while the boys did manly events like rings, pommel horse, and parallel bars.

Yeah we had pommel horse and parallel bars too but not rings. Rings would be the real hard one for sure.

Thanks. I used R (R Project for Statistical Computing) to plot the curves.

Definitely it would be hard to prevent angle error from increasing with velocity...

I'm curious how DS wanted the problem to be interpreted; usually he emphasizes problems that can be solved elegantly by noticing some symmetry or invariance, etc. But here I don't see anything like that?

I didn't do a good job of phrasing the problem. Among other changes I would make would be to make the ball a mere point and make the diameter of the basket one foot.

Lol. Some other human factors engineering problems:

The average basketball player weighs around 225 lbs. and the basketball weighs 1.375 lbs. Newton's Third Law applies. This might be important if we would like to take more than one shot. "Tom Cowley has just left the building."

Humans cannot jump infinitely high. In fact, we can't jump high enough to take a straight shot unless we are very close to the basket. We can safely limit the release height to under 15 feet since no one can even reach that high.



Humans are not infinitely strong. We can safely limit initial velocity to 1.5 feet per second, since no one can throw a basketball with that.



And a trigonometry problem:

As your height of release increases, the ellipse you are aiming at becomes more circular at the entry (positively influencing theta error tolerance since we have to take into account that the basketball is not a point) BUT at some point the extra distance is making more of a negative contribution theta error tolerance than the increased circularity is positively contributing.

Up until your release height (assuming that we just let you climb on a box to avoid all the human factors engineering stuff) is equal to the height of the rim, they work together at any given theta. The minimum best height of release (again, ignoring human factors) is at least equal to rim height.

No one does granny shots anymore. At least no one at the professional level. I was very horrible at basketball and enjoyed the story. One of the guys I used to play soccer with in grad school moved to Greece to play pro basketball. I can't remember his name, but he was far too short to play professionally here. He was an amazing soccer player and all around good dude. "Brian, I know you dislocated your right shoulder last week. I will make them shoot to your high left today so you can rest it."

Also, lol that DS has me thinking about this (exercise that brain, Brian!). My intuition is that we get the best results (in terms of both theta and velocity tolerance, just for swish shots) by finding the minimum possible parabolic arc length divided by the length of the ellipse we need to hit. I am fairly certain that such a thing can be graphed given hoop width, ball width, distance (and of course gravity).

Rick Barry used a granny shot, and he's 3rd place for all-time career free throw percentage with 89.98%, with first place being Steve Nash with 90.41% (Mark Price is second with 90.39%). Barry once went 94.7% for a season. He said he could shoot 80% with his eyes closed.

Physics Proves It: Everyone Should Shoot Granny Style

I haven't checked that guy's trigonometry for optimal angle for a given height.