No Calculus Needed?! How to Maximize Range Using Simple Geometry.

Рет қаралды 956,491

3 жыл бұрын

This physics problem, determining the launch angle that gives the maximum range when the starting point is above the level ground, is a personal favorite since it took me a year of cogitating to come up with this solution. No, I never threw anything off a building, but I heard about some rowdies who did, and that is what got me to thinking about this problem.
Of course, ignoring air resistance and other complicating factors means this is just an approximation (and quite possibly a bad one), but accuracy not the point. Such simplifications are useful for teaching the basic concepts of physics and geometry, and for the simple pleasures of exploring the mathematics involved.
BTW, please do not throw things off of buildings! Also, if you throw anything into a body of water, make sure it's natural, not trash.
The method I came up with to solve it involves no calculus or quadratic equations. All you need is elementary physics (conservation of energy, plus basic velocity vector concepts) and some elementary geometry (area of a triangle, simple trig functions). I can't be the only person to come up this solution, but I have have never found anyone who has seen it before. All the websites and videos about this problem use calculus and complicated trig identities. And none of them point out that the denominator in the answer is the final velocity of the projectile. If anyone has heard of this solution, please let me know in a comment.
Update: Commenters have pointed out that this solution is well known by Russian Physics Olympiad competitors.
Animations were created using Desmos Graphing Calculator (www.desmos.com/calculator) with the help of DesmosPlayer, a browser plug-in that I wrote to control the graph (github.com/MathyJaphy/DesmosP...) Screen capture turned that into a video; edited with iMovie.
Music: “Off Broadway”, iMovie Song ( • Off Broadway | iMovie ... )
Corrections:
1:35 “Ray” is the wrong term here. Velocity is represented by a “vector”. A ray doesn’t have a specific length. It extends infinitely in one direction.

Пікірлер: 1 200

@SlimThrull 2 жыл бұрын

"Whatever you can solve with calculus, I can approximate with algebra."

@SoloRenegade 2 жыл бұрын

mere geometry and sketches in CAD can also solve otherwise complex calculus problems too.

@adb012 2 жыл бұрын

The cool stuff here is that this in not an approximation, but an exact solution.

@SlimThrull 2 жыл бұрын

@@adb012 Oh, don't get me wrong. This is indeed a very cool solution. A little bit of trig and some algebra and, bam, you get the exact answer. I just have a personal vendetta against Calc. I use any other method to solve a problem out if sheer spite. 😉

@promethius7820 2 жыл бұрын

When you realize that one form of calculus is the essence of testing a series of better approximations until, for all rights and purposes, the approximation is the actual value.

@sethdon1100 2 жыл бұрын

Newton’s method goes brrr

@adityabhandari271 2 жыл бұрын

I had asked a question on Physics Stack Exchange, "Why does the optimal angle depend on velocity?" Where someone answered (then suddenly deleted) and told me to check out this video. And this is a realllllllllllllllllllyyyyyy coooooooollllllll video, really nice solution. Thank you!

@MathyJaphy 2 жыл бұрын

Hmmm, you're the second commenter today who's mentioned Physics Stack Exchange. I should check it out. Strange that the reply was deleted. I wonder if there are bots that delete posts with KZfaq links in them.

@adityabhandari271 2 жыл бұрын

@@MathyJaphy yes, they might have a bot like that I think (and for good reason lol). I posted the same comment twice just one without the link coz I thought this was going to happen. again, awesome video!

@nikhilnagaria2672 2 жыл бұрын

@@MathyJaphy they delete it because SE has a wonderful policy. (No sarcasm, it's good for real). This is because just posting a link adds nothing as an answer, either it should be written in comments, or if written in an answer, the link should be complimentary and not the answer itself.

@thetrickster9885 2 жыл бұрын

Bro you in 11th?

@terdragontra8900 2 жыл бұрын

The angle must be constant with respect to the ratio of the throw velocity and the height of the building (because you can scale up a trajectory into another by scaling height and velocity up but keeping gravity constant), therefore if you believe the angle isnt constant for all starting conditions (which is intuitively clear), it cant be constant with respect to building height alone

@tsawy6 2 жыл бұрын

I've tried to answer this question at least 4 times in my life, with a focus on the intuitively obvious but interesting fact that the launch angle approaches 0° as the height increases. I think I managed that small problem, but frankly it wasn't very rigourous. This is grand!

@melancholiac 2 жыл бұрын

You and me the same. Once we are sufficiently high up, the flat launch becomes best.

@CadillacDriver 2 жыл бұрын

You have? This has been around since maths itself.

@nightsout. 2 жыл бұрын

What I like about the end result is that it still works for the H=0 case (when you're on the ground), and you still get 45 degrees! Great video!

@mschauer97 2 жыл бұрын

I mean if that wasn't going to work for H=0, the formula would have been proven wrong..

@giusepesm 2 жыл бұрын

Does it work for negative values? (When you're below ground level that rock will land on - on a ditch for example)

@5014eric 2 жыл бұрын

Yes, it does. Around age 15-16 I worked out iirc, 48 deg was optimum for hitting the ball over a fence 4m higher than the start point 30m away

@terdragontra8900 2 жыл бұрын

@@giusepesm yes, but the formula for final velocity has no solution (square root of a negative) if the height is too negative (meaning even throwing straight up wont reach ground level)

@klaasbil8459 2 жыл бұрын

I did that mental check as well. A tower with height equal to 0 is just a special case of a tower of height h.

@shiinzshiro4447 2 жыл бұрын

I love how geometry has a solution to everything. Triangles, triangles everywhere I look

@LeavingGoose046 2 жыл бұрын

Truly the apex of math

@Ferdaev 2 жыл бұрын

Yeah. Almost everything in math is useless without geometry

@LeavingGoose046 2 жыл бұрын

@Coup Lab Geometry is the easiest way of using imaginary numbers though? All the numbers become a grid lol

@solapowsj25 2 жыл бұрын

And the physics, which reduced the event to a straight line➖ where both horizontal and vertical vectors were proved to be constant.

@vishalmishra3046 2 жыл бұрын

In India, under CBSE board, the Physics NCERT books had (and perhaps still do) what are called star-problems. They are unusually hard problems not meant for everyone but only meant for students who are pretty confident about their physics and math skills (in this context - classical mechanics and differential calculus skills). I ran into this problem for the first time few decades ago and ended up solving it the hard way. Happy to see that there exists a simple and elegant solution for this seemingly easy but actually a pretty hard problem !!

@MathyJaphy 2 жыл бұрын

Thank you for taking the time to tell me about this.

@ozzymandius666 2 жыл бұрын

@@MathyJaphy So, the higher you are, the smaller the angle, in a simple 1g field. You have told me nothing yet. I'll have to a bit of math to figure out max distance angle as a function of height, which you failed to give.

@jackm.1628 2 жыл бұрын

So you found a root of the derivative? How?

@TheKai190 2 жыл бұрын

@@ozzymandius666 6:25 with the height and your throwing speed known (assuming its constant as said in video), you can first calculate the landing speed and with both speeds then the throw angle for max distance. So only step that is missing is to combine the 2 formulas if you want to have it in one step!

@borat1 2 жыл бұрын

@@MathyJaphy this method is actually genius. I am going to use this in a computer program.

@andrewwilmot718 3 жыл бұрын

I remember a certain person posing this exact question 30-some odd years ago in a work office in Bethesda. Nice analysis!

@MathyJaphy 3 жыл бұрын

That’s the office where I finally figured out the answer during a boring meeting in the conference room. I don’t remember talking about it, but I was so pleased with myself I suppose I must have. I’m impressed that you remember.

@anjugour9295 2 жыл бұрын

@@MathyJaphy Sir I got the expression to find the maximum range that is, arccos {√(2gh+v²)/√(2gh+2v²)} where, h is the height of the building v is the initial velocity

@MathyJaphy 2 жыл бұрын

One thing missing from the video's narrative is that our little stick figure is throwing as hard as it can. It wouldn't be a very interesting problem if it could "just throw harder", as so many commenters have suggested. :-)

@SirRanjid 2 жыл бұрын

Yeah then take a stronger stick figure.

@rashiro7262 2 жыл бұрын

For those who still have trouble understanding the solution, you can think of is this way: You need horizontal speed to drive the ball forward and airtime, so it has time to travel. By aiming higher up you're essentially trading your horizontal speed for airtime. If you're throwing from a roof of a building you already have x amount of guaranteed airtime due to the height, so you need to trade less of the forward speed. Therefore (with constant throwing speed) the higher you're throwing from the closer the angle will be to 0°. On the other hand, at 0 m height you have 0 guaranteed airtime so you have to trade the throwing speed equally, which you can achieve by aiming at 45°. This is due to the simple fact that at 45° the horizontal and vertical components of your throwing speed vector will be equal (sin 45° * v0 = cos 45° * v0).

@andrijacvjetkovic4662 2 жыл бұрын

With this formula you can calculate perfect angle to get out of hole amaizing i must say

@backyard282 2 жыл бұрын

What a genius solution! I would've gone brute force calculus and then get stuck trying to solve for the angle

@NexusEight 2 жыл бұрын

Good stuff MJ. Never thought to have pondered further beyond the "45degrees is the best angle to kick/throw a ball". Luckily I came across this video because not only did you answer a question with great intuition, but you also, and more importantly, thought to have asked a terrific question. Great Job!

@empty5013 2 жыл бұрын

this is actually super useful for calculating maximum trajectories when developing a video game, since most video games ignore air resistance for projectiles. Technically you can just simulate the whole projectiles path, but this is more efficient and would take way less code, awesome!

@bocchi2403 2 жыл бұрын

Wait what, game programmer need to study physics? The game physics is derived from real physic coded into the game?

@aty4282 2 жыл бұрын

@@bocchi2403 pretty much, yes

@amazuri3069 2 жыл бұрын

@@bocchi2403 how do you expect games to look and feel realistic then?

@samuelbuckner7599 2 жыл бұрын

Actually, you wouldn't need to conduct a series of simulations even if this elegant solution did not exist -- this problem can also be solved numerically by applying a root-finding technique to the equation presented at 1:09.

@asdfggfdsasdfg3762 2 жыл бұрын

@@bocchi2403 You can also use build in physics engines if you use game engine like Unreal Engine, Unity, etc... Usually people do it that way if they use game engine, but sometimes its better to do yourself, if you dont need to do very complicated stuff and need to do something unusual

@KazeN64 2 жыл бұрын

This is really really cool, but a few times you've treated the length of a vector to be the same as the vector itself. That obfuscated the fact that we still don't have an actual formula that can be solved exactly by plugging in the variables at the end here. Would be cool to investigate this a bit more.

@TerraBlo 2 жыл бұрын

obfuscated

@joalampela8612 2 жыл бұрын

@@TerraBlo Yes, obfuscated. Perspicuous as the daylight sky!

@TerraBlo 2 жыл бұрын

Obfuscated 2

@BrandNewByxor 2 жыл бұрын

you look absolutely ripped in a tank top bro

@Epyxoid 2 жыл бұрын

I like the fact as a non native speaker, that there are English words even native speakers despise xD Love it!

@secretagent86 2 жыл бұрын

i must be always standing on the ground as this presentation is way over my head

@justinblackwood4241 2 жыл бұрын

If you're standing on the ground then be sure to throw the small projectile at a 45 degree angle 😂

@idirkhial9422 2 жыл бұрын

Wow! For a long time I’ve been trying to find an intuitive way to figure out the 45 deg optimal angle without the equations of motion... and now I’ve stumbled upon a general method for all heights! Thanks a lot!

@RannyBergamotte 3 жыл бұрын

My mind is absolutely blown. This is so elegant and wonderful. Keep up the good work!

@TheLaxOne 2 жыл бұрын

I’m glad the KZfaq algorithm showed me this video, since I’ve been trying to find an elegant way to solve this problem to explain to my sister without needing calculus. Excellent video!

@not_vinkami 2 жыл бұрын

My physics teacher just finished the projectile motion chapter, and you concluded everything in it in 7 minutes

@JanxakaJX 2 жыл бұрын

Incredible video. Really happy that you're continuing to make videos too, 5 months later.

@jamesorendorff2284 2 жыл бұрын

I just watched three of your videos consecutively, each more unique and nerdy than the last. This is beautiful stuff, thank you for making it!

@WindMills_ 2 жыл бұрын

This was really good video. All the points presented are so intuitive that my mind was in awe. Thanks and got subscribed :D.

@glashoppah 2 жыл бұрын

Beautiful. Analytic geometry has solved a lot of things in front of my eyes that looked like they were going to take a lot more work using algebra or calculus.

@justacat2318 2 жыл бұрын

This is more interesting than my whole year of physics class

@holdenmatheson2185 2 жыл бұрын

This problem can be re-imagined as maximizing the parabola, which happens when the hypotenuse of your triangle passes through the parabola's focal point. This then necessitates that the starting and ending points are 90 degrees offset.

@grantdraus7449 2 жыл бұрын

That's a solid idea

@BillBrasky368 2 жыл бұрын

Back to your crayons. Haha, As an army vet I just can’t accept that this intelligent comment came from a Marine.

@holdenmatheson2185 2 жыл бұрын

@@BillBrasky368 That's the trick: the purple ones make you smarter.

@noneofyourbusiness4735 2 жыл бұрын

What amazes me much more is that you go from the roof to your apartment, only to be in both places at once a short time later. Wow, how did you manage that? Awesome!!!

@MathyJaphy 2 жыл бұрын

Haha. He's just imagining himself on the roof. The real guy is solid black, and the imaginary guy is faded to grey. I hoped that would clarify that there's no disconnect in the time-space continuum.

@amalantony8594 2 жыл бұрын

Wow, I tried to find the optimum angle of throw for a javelin thrower considering his height and average velocity of the throw when it is released from hand. And I was struggling to find the optimum angle and gave up. This happened at the time of Olympics, and now youtube decides to recommend this video. Anyway I'm pleasantly surprised.

@techdoc99 2 жыл бұрын

Very cool approach to this ubiquitous physics problem! And also very well presented!

@brandonjohnson735 2 жыл бұрын

Awesome video! I haven’t interacted with calc or physics in the past few years but this was a great way to stretch out that part of my brain again, thank you!

@damiensadventure 2 жыл бұрын

Okay you got me with the guy sneaking off at the beginning. I've already learned this. Even though my haven't. But I found myself stuck watching lol. Good video :)

@dhruvaggarwal4461 2 жыл бұрын

This is wonderful. the final velocity being perpendicular to initial velocity occurs in many similar problems. We had done this in class using vectors which gives the same conclusion ( u can find the method in 200 more puzzling problems) but the area of a triangle is a great way to look at things and can be understood by those who might not have studied by vectors yet.

@kukaracila2152 2 жыл бұрын

I didn’t understand a damn thing, but I like your funny words magic man.

@sukhjinderkumar2723 2 жыл бұрын

great video , keep going, it was fun, i remember doing calculus when asked by prof. and he just left the problem saying maths is too tough. And today i finally have a clean and intutive way to look at it

@chess1011 2 жыл бұрын

It was uploaded 6 months ago, and I'm finding it now. KZfaq, why you never recommended this channel before! A very unique solution indeed.

@sergarlantyrell7847 2 жыл бұрын

That's the physics answer... Assuming it's a spherical cow in a vacuum... The Engineering answer is to exploit the magnus effect to generate lift from the ball in the direction you want the ball to travel, thereby "throwing" it further.

@alan133 2 жыл бұрын

The Computer Science answer is to write a simulator and run it until it is 99.99% sure its the perfect angle.

@truthseeker7815 2 жыл бұрын

Biology answer?

@catchara1496 2 жыл бұрын

@@truthseeker7815 adrenaline

@truthseeker7815 2 жыл бұрын

@@catchara1496, oh yeah, Fight or Flight State

@aeaeeaoiauea 2 жыл бұрын

@@catchara1496 Philosophy answer?

@sigisalmen2399 2 жыл бұрын

I'm more the practical guy. I would do it with a garden hose and an angle gauge. Turn the on water and change the angle till the water hits the ground at the farest point. Then I'd measure the angle of the hose at it's end. No calculus needed. In the worst case, someone gets wet. 🗣️"SORRY, THAT'S JUST PHYSICAL EXPERIMENT!"

@elijahdschultz 2 жыл бұрын

This approach only works if you throw the ball at the same speed that water leaves the hose.

@sigisalmen2399 2 жыл бұрын

@@elijahdschultz If I want to know the distance, yes. But it's about to find the optimal angle. And that works even if I scale it all down. Doesn't it?

@elijahdschultz 2 жыл бұрын

@@sigisalmen2399 unfortunately no. As the video demonstrates, the optimal angle depends on how fast you can throw.

@sigisalmen2399 2 жыл бұрын

@@elijahdschultz I guess now I can't avoid to get on the rooftop and make some passing stranger wet. Have a nice day

@yuttor0013 2 жыл бұрын

@@elijahdschultz You could also use the knowledge gathered from the water's course to approximate how a different projectile of a different speed would fly with no math necessary.

@bvwalker1 2 жыл бұрын

Wow! This is a great problem with a fantastic solution. Thank you for sharing. Subscribed!

@scraps7624 2 жыл бұрын

This was amazing dude, great work!

@adb012 2 жыл бұрын

Wow. Engineer and Physics teacher here. That solution is SO creative, beautiful and simple (once you know it). The real amazing stuff here is not the solution itself, but.... HOW ON EARTH DID YOU COME UP WITH IT????? This is almost a proof that P ≠ NP. This is the first video that I watch from you and before even going to your channel to see what's else in there, I liked and subscribed.

@MathyJaphy 2 жыл бұрын

Thank you for your kind words. I was stumped by the derivative equation, so I started working out related formulas on a calculator and plugging numbers in for fun. When you do that long enough you sometimes notice patterns, and I just got lucky. I worked out the formula for the final velocity's angle, and when I plugged in some numbers, I noticed that the maximum distance occurred when the velocity angle changed by 90 degrees. With that as a conjecture, I figured out the final formula. Then it took me a long, long time to figure out how to prove that conjecture. It seems so simple in retrospect.

@Kiba114 2 жыл бұрын

no teacher writes like this

@adb012 2 жыл бұрын

@@Kiba114 What do you mean?

@umershaikh8012 2 жыл бұрын

@@Kiba114 ?

@farhanaditya2647 2 жыл бұрын

@@Kiba114 wut?

@elton8135 2 жыл бұрын

wonderful animations and wonderful explanation, i love the fact that you mention that this problem assumes negligible curvature of earth xD but that does make me wonder, what if...

@MathyJaphy 2 жыл бұрын

Ooh! Hold my beer…

@llzz1528 2 жыл бұрын

It is

@prototypeinheritance515 2 жыл бұрын

if the earths curvature was modelled, the solutions wouldn't behave nicely. I mean you could throw it so fast it would orbit forever

@jackm.1628 2 жыл бұрын

@@prototypeinheritance515 Wow that's a cool observation. A discontinuous function arising from physics. I haven't seen such a thing before.

@junkgum 2 жыл бұрын

And the Jeopardy answer is: What is a satellite?

@gdevelek 2 жыл бұрын

You made quite a leap of faith, going from distance to area of triangle. No math or physics professor would let you get away with it.

@MathyJaphy 2 жыл бұрын

A leap, yes, but hardly one of faith. :-) It's a perfectly valid observation, and therefore acceptable in a proof. I expect a professor would encourage exploring the reasons for the connection, which I omitted from the video (since I don't fully understand it).

@gdevelek 2 жыл бұрын

@@MathyJaphy Christians also don't fully understand their god, yet they have FAITH in him. I rest my case.

@Sam-gv9tv 2 жыл бұрын

This is amazing! Keep up the good work :)

@jayteach6787 3 жыл бұрын

This is great! Like 3Blue1Brown but with Desmos. Could you post as a reply the Desmos graph you used to make this?

@MathyJaphy 3 жыл бұрын

Thanks! I'm honored by the comparison to 3b1b. The graph for this is a mess. I made the video with a javascript application that I wrote to control all the changes, so it didn't have to be very well organized. Nevertheless, here is a link to the graph: www.desmos.com/calculator/97tdosccov If you care to check out my three other videos, you'll see they all have a link to their more reasonable Desmos graphs in the description.

@droro8197 2 жыл бұрын

I actually never thought of this problem further than the case of "same level" optimal distance . Very nice! Shame you dont publish more often (yes, I know its hard)

@smartproject5614 2 жыл бұрын

Excellent presentation! And very thorough. Thanks, MJ.

@juan.s 2 жыл бұрын

really well done with the animations and everything, i thought the channel would have a lot more subs!!!

@dougsteel7414 2 жыл бұрын

What I find interesting about this is that animals (including us, though I'm sure we're getting worse) make these calculations in an unfathomable parallel fashion, and once they've learned the responses of their muscular system and visual ranging seem to get it very accurately. It's plausible there's a time-like differential electrochemical relationship in the nervous system that models/reflects it. Kind of nonsense I dream up at the top of buildings

@Orakwan 2 жыл бұрын

That would be interesting to study deeper. Animals have this sort of instinct, but how, why, when? We exist today because this has been a useful genetic advantage at some point in the development of the species, and as you say it's getting worse, probably because we don't need it for survival anymore, though we like using it in many sports

@dougsteel7414 2 жыл бұрын

@@Orakwan it would be fascinating. There might be a clue to some aspect of how and when mammals in particular seem to by a mysterious mechanism "know" how to move at birth; it's generally believed humans are born in a sense too early, at an underdeveloped point, in order that our skull volume doesn't make birth mechanically impossible or harmful. Horses etc. seem able to walk almost immediately. Presuming actual knowledge isn't inherited, some kind of system must be, it could have implications for the way robots are designed. Even the most advanced ones are incapable of walking in the way living creatures do, with that level of dexterity.

@surajvishwakarma4534 2 жыл бұрын

Dude.. you're underrated

@gauthierruberti8065 2 жыл бұрын

You just can't imagine how much I needed this!

@tedsheridan8725 2 жыл бұрын

I'm very surprised I never came across this version of the problem before. Very cool solution!!

@thf1933 2 жыл бұрын

Omg thank you so much I spent whole day yesterday trying to throw a stone to the damn river till neigbours got to the roof and kicked my ass!

@gw819 2 жыл бұрын

But, how do you know the final velocity? Yes it is gravity*time, but how do you know the time component? Since the time component is dependent to the maximum height, among other things, which is dependent on the throwing angle. Edit: Nevermind! re-watched 3:45

@FareSkwareGamesFSG 2 жыл бұрын

The first minute you described, I did *EXACTLY*. Reached a dead end, even with WolframAlpha. I'm dying 😂

@ThePopeOfAllDope 2 жыл бұрын

This is some prime content. Very beautifully done.

@hansmustermann4986 2 жыл бұрын

This is some great deduction to a nice question! It still leaves me wondering though: What is the optimal angle? I have no use for arctan of V0 over Vf. I don't know my V0. It would have been nice to see a diagram for the angle against the height for some standard V0 for example to get some intuition.

@Goku_is_my_idol 2 жыл бұрын

So here's my take on this: The angle is calculated as arctan(V0/Vf). V0/Vf=1/√(1+2gH/V0²) . Now we need a relationship between V0² and 2gH (without actually knowing V0). Stand near the tall building and throw the stone vertically upwards as hard as you can. Estimate how close the stone gets to the top of the building. Lets take the height as xH (as a multiple of building height). From energy conservation V0²=2gxH V0²/2gH = x Throw the stone from the building at the same velocity (since we cant be sure if the velocity is going to be same i would recommend throwing it as hard as you can both times) at the angle arctan[√(x/1+x)] to get the maximum range. This is totally impractical but just wanted to share something lol.

@tedsheridan8725 2 жыл бұрын

@@Goku_is_my_idol I worked it out similarly - except I have tan(theta) = root(x/(1+x). I actually did it in terms of the reciprocal, K = 1/x. Then tan(theta) = 1/root(1+k).

@Goku_is_my_idol 2 жыл бұрын

@@tedsheridan8725 yes you're right It should be arctan[√(x/1+x)] I did it in my head while writing the comment hence the miscalculation

@YourMJK 2 жыл бұрын

1. Determine height of building: Drop a stone and measure the time it takes to land. h = 1/2gt² = 4.9m/s² * t² 2. Throw a stone as hard as you can at 0° and measure the distance on the ground. If you don't want to go downstairs for the measurement, use a protractor to look at the landing spot, measure the angle and use d = tan(θ) * h. 3. Calculate V₀: t = sqrt(2h/g) (or just use t from 1.) V₀ = d/t Now you can plug in your V₀ into the arctan equation from the video and get your optimal angle.

@emperorsascharoni9577 2 жыл бұрын

I did not understand, but still was fun watching. This is like watching Lee Sedol vs AlphaGo when you dont even know more than the basic rules of the game.

@royck5646 Жыл бұрын

MJ Thank you for your very elegant post two years ago, that I only recently came across

@MagnusWendt 2 жыл бұрын

Yes! This video was for me. Thank you! In jr high school my PE teacher told me to use a 50deg launch angle when shot putting and while that didn't sound right there are anatomical considerations to make and he wasn't my math teacher so I didn't think much of it at the time. The launch angle question stuck in my mind though, but not so much that I took the time to dig in to it. It pleases me that it has such an elegant geometrical solution. :)

@factChecker01 2 жыл бұрын

Is this a different answer from what you would get by imagining the path as part of the path of a 45-degree throw from ground level? Can't you just figure the 45-degree path of a throw from ground level that goes through your current position and use the angle from that partial path?

@MathyJaphy 2 жыл бұрын

Nope. Firstly, a launch from a negative x-coordinate on the ground at a 45-degree angle would maximize the distance from that starting point, but it would not maximize the distance from the real starting point, even if the path goes through the real starting point at the same velocity. Further, any imagined launch from the ground at a 45-degree angle would hit the ground at the same 45-degree angle. But the path of maximum range from the real starting point does not. There's probably a better way to explain it. Maybe someone else reading this comment could give it a try.

@factChecker01 2 жыл бұрын

@@MathyJaphy , Good point. I stand corrected.

@erikanybody4298 2 жыл бұрын

@@MathyJaphy are you sure? The path the stone takes from the top of the building to the river must be a part of a parabola. If we say the building is at (a,b), and the river is (n,0) and you're trying to maximize n-a distance, where both (a,b) and (n,0) are on a parabola, I can't see how this ISNT the same parabola which passes through (0,0) and has n maximal. Granted the velocity thrown at (0,0)(called V1) is not the velocity at (a,b) (called V2). But there's only a single parabola that has V2 at (a,b) and n-a maximal. This is the exact same parabola that has V1 at (0,0) and -V1 at (n,0) with n maximal. E.g. the parabola of 45 degree tangent at (0,0) AND -45 at (n,0). Problem, of course, is that we don't immediately know what (a) is. We know (b) and we know (n-a) and we know V2. I'd have to do more math, but I can't see how the angle when the stone hits the river is NOT 45 degrees when maximizing the distance from the building for a given thrown velocity. That is, we're trying to solve the problem of "if I can throw a stone at X velocity from a building Y high, how far away can the river be and I still hit it?" The answer will be the angle thrown for the max distance.. e.g M degrees to get Q feet. We're NOT solving the problem of "the river is X distance away and I'm Y high on a building, what velocity do I have to throw at to make it?" This question doesn't have a single answer, because it will be a whole bunch of angle + velocity pairs.

@MathyJaphy 2 жыл бұрын

@@erikanybody4298 Think about the extreme case where the initial speed (at the top of the building) is small compared to the height. There’s no launch angle that could result in a 45 degree landing angle, yet we can still ask which one maximizes the distance.

@scipio764 2 жыл бұрын

Ah yes, I remember doing this during my Freshman year in HS. This one question amounted to 0.5/10 points. No, it was not easy; but the policy is that the harder the question the fewer points it provided, because hard questions just serves as a cut to separate better student from the rest.

@robj144 2 жыл бұрын

Maximum when range when vertical displacement was not zero was on your Freshman HS school test???

@usptact 2 жыл бұрын

Brilliant presentation and explanation! Great work!

@timdawn705 2 жыл бұрын

I’m so glad this video exists. I use to completely not even understand how to throw rocks from a building at a perfect angle, and now I still don’t.

@garrybowers9998 2 жыл бұрын

This was an interesting view on an age old problem. I've always considered approximately 37.5 degrees to be optimal when including air resistance, not too far off from you (whereas you didn't factor in any air resistance).

@jk23233 2 жыл бұрын

How did you get a specific optimal angle of 37.5 degrees? What angle would it be if you ignore air resistance? Does your answer not depend on the initial velocity Vo and the height H?

@dannywhite132 2 жыл бұрын

If I had a pound for every time i heard "assuming ideal condition" I wouldn't have to be at university right now

@atalazs 2 жыл бұрын

The ability to idealize and approximate and still get a useful result is the most important skill of a physicist.

@dannywhite132 2 жыл бұрын

@@atalazs OK....?

@Frank_Lee_Terrible 2 жыл бұрын

Such a great video! Intuitively I came up with the same answer but had no idea why.

@solypsomancer9540 2 жыл бұрын

I derived this formula in 1990 when playing a WWII submarine combat simulator. Firing angle to target to hit perpendicularl amidships was invtan t° based on current bearing to target if my velocity was zero. Projectile and target velocities were constants that "fell out" of the equAtion if i, the firing platform, was stationary. It made the game trivial. Thank you for reinforcing my assumptions from 30 years ago in this video. Same problem effectively. Different visualization. It made me vizualize the basis of the analog targeting computers made during the war.

@uncommonsense360 2 жыл бұрын

can't you just run a simulation with a range of launch angles, plugging in your building height, see which one goes farthest and then throwing at that angle?

@kito323 2 жыл бұрын

Sure thing but i think the problem here is that you might not be carrying a PC and have that time to write/run scripts every time you wanna throw rocks. The idea would use some simple solution that can be solved at max with your phone calculator. But I guess you can also write the script to work as a phone app too so yeah...

@RedHorseArcher 2 жыл бұрын

Теоретик: я бросил камень недостаточно сильно, а значит - необходимо потратить много часов на высчитывание вектора скорости и идеального угла для броска! Практик: Ладно, просто попробую снова.

@cagedgandalf3472 2 жыл бұрын

This is amazing I've thought about this problem during physics what would be the optimal angle in a throw because it's always been given by the problem. And this video answered my question, physics is cool.

@user-ko8pu5wu4l 2 жыл бұрын

I will come back to this video when I have learned more about physics

@Nikioko 2 жыл бұрын

45°. Easy. Artillery pieces with lower angles are called cannons, pieces with higher angles mortars. And a howitzer shouts in both angle groups.

@thorr18BEM 2 жыл бұрын

Apparently you didn't watch the video or you'd know it's not 45°.

@Nikioko 2 жыл бұрын

@@thorr18BEM Of course I didn't watch the video. What sense does it make to solve a puzzle after seeing the solution?

@adiaphoros6842 2 жыл бұрын

@@Nikioko So you don’t even know the question to begin with.

@davidbrisbane7206 2 жыл бұрын

Cool.

@MrMawoolf 3 жыл бұрын

Very woke not to have used the old "muzzle velocity" metaphor.

@xadxtya 6 ай бұрын

Never been taught such an elegant solution, especially without calculus.

@moonman2183 2 жыл бұрын

So many aha moments with this one, which is the best feeling! Thank you for making this

@bobingstern4448 2 жыл бұрын

This is epic! My first thought was to use newtons method to approximate the the angle using the derivative but this is so much simpler!

@Fred-yq3fs 2 жыл бұрын

I had the idea of using air time and playing with velocity vectors but did not push it far. You kept at it for 1 year. I'm in awe. This is absolutely brilliant, well done!

@MathyJaphy 2 жыл бұрын

Well, a year of thinking about it now and then. It wouldn't let go of me! Glad you enjoyed it. Thanks for the kind words.

@deedatfatahillah7164 2 жыл бұрын

Omg please upload some more videos like this i absolutely love it

@RishabhSharma10225 2 жыл бұрын

This video has strong 3b1b and minutephysics vibes. Way to go man!

@cdab305 2 жыл бұрын

Good job! Clear and elegant!

@Michaelfirefoxx 2 жыл бұрын

Gotta love watching a video with massive helpful knowledge in it and having understanded nothing from it.

@adamant8435 2 жыл бұрын

I literally tried solving this problem with calculus with my classmate today and when I came home this video appeared!

@alaspooryorick9946 2 жыл бұрын

Nice! I think about this every time I water the garden. Right on

@MbKTheGLow 2 жыл бұрын

I'm glad this video had pretty pictures, otherwise I would have been really confused.

@user-db7ru9cd2d 2 жыл бұрын

Wow!! That was amazing!

@wayneyadams 2 жыл бұрын

The final velocity equation can also be determined using kinematics equations for free falling bodies (that means we ignore air restistance).

@crownclown1951 2 жыл бұрын

The power of triangle, nice illustration btw want some more like this video. Up.

@ioanstef1983 2 жыл бұрын

Thank you for this nice video, Well done!

@oadka 2 жыл бұрын

Such genius. Absolutely love the video

@marclanman1902 2 жыл бұрын

In an alternate universe, I can understand this. That makes me happy.

2 жыл бұрын

I'm so happy I discovered this channel! Next to Mathologer and 2blue1brown, this is exactly what I enjoy. :)

@Alg007 2 жыл бұрын

I swear next time I am on top of a building I will never ever think of throwing anything… this has exhausted my brain !!

@MathyJaphy 2 жыл бұрын

:-) kzfaq.info/get/bejne/p7eIppVorZ3SYYk.html

@cheese3038 2 жыл бұрын

very elegant solution i probably would have gone with the hard way if i had to do this problem.

@johnquest3102 2 жыл бұрын

VERY NICE!! great analysis, me likey!

@millionelectricvolts6117 2 жыл бұрын

Good presentation man, im gonna sub for more 😄

@shirish11 2 жыл бұрын

Good one. Never thought of that.

@eitanzeevi1894 2 жыл бұрын

When you major in computer science and just run a simulation in python for every angle accurate to 0.001 degrees to find the best one:

@sandrogiorkhelidze1359 2 жыл бұрын

very good video i for sure expected you to have 2mil subs

@TheSpiffyNeoStar 2 жыл бұрын

This is the approach I've been looking for to solve my age old question of "what is the ideal angle to jump off of a swing to maximize horizontal distance". Time to bust out the ti-83 and get crunching some numbers, because velocity leaving a swing isn't constant based on angle, but this should get me closer than my past brute force methods.