Hope you guys enjoy! As stated at the end I'll taking a little holiday break so will return in the New Year (possibly with a new skit video). You can check out the second channel ruclips.net/user/zachstarhimself for exclusively skit videos and as always check out STEMerch.com to get the floating globe, meme shirts, and more.
That was awesome but the first one is utterly counterintuitive and I think needs a video of its own...really...shouldn't it travel fast enough in the forward direction due to the same velocity of wind taking it in the first direction to make up for the slowing caused in the second direction...the first direction might almost be instantaneous whereas second one takes the amount of time taken by the no wind plane to go back and forth?...please explain...thanks a lot.
And pretend the cosmic microwave background doesn't exist, and neither does heat transfer by radiation. Oh yes, and all collisions are perfectly elastic
David. Interesting thought! Why do you think this? It's certainly an attractive idea because you start your spiraling earlier. but the spiraling isn't the essence of what you're about, surely. You want to find the other boat earlier and since 50% 0f the time the spiral is away from the target, no matter which way it chose to flee, therefore 50% of the time your early spiral is giving the target extra time in which to get more distant, surely? A thought which may or may not be equivalent, I'm not sure, in 50% of all cases starting early to veer away from your path straight ahead on known information is adding to the time the target has to add to its distance from the last place you saw it, don't you think?
@@lordfnord5768 if one thinks in worst cases, my method is better. Once you start spiraling, you have to max do one round. Since I both use less distance to where I start and have a smaller spiral (the chased boat haven't gotten as far when I reach it) it will be faster.
@@lordfnord5768 also, in the case in the video the 50% who turn backwards make that spiral longer. So it is the same either case, but i travel less before starting spiraling.
Often this reminds that our 4-5 grade mathsbooks contained more word problems involving real life question , but in 9 grade chapters word problems involving real problems are as rare as Halley's comet
the way I saw the first one is you'll be slowed down for more time than you'll be sped up, so the slowing down is more effective and it'll be slower on the whole.
6:55 : I have discovered a truly remarkable proof of the irrationality of pi plus e, but the margin of this RUclips comment is too small to contain Fermat's last theorem.
My intuition for the first puzzle is similar. If we consider a headwind with equal velocity to the plane's, clearly the return trip takes an infinite amount of time. Therefore, it can't always be equal.
Suppose wind flowing in direction a to b now at first plane will reach twice fast but when returning as wind speed against plane is same , the plane velocity will be 0 and plane won't move which ultimately makes time infinite 🥱.
"I don't need any math, I can think." -- Joan Robinson, English economist. The wind is going to slow you down one way. Going slower you will be slowed down for a long time. The wind is going to speed you up going the other way. Speeded up you are going to be speeded up for a short time. You're speeded up by the same speed as you're slowed down by -- but there's more slowing down than speeding up going on, so you'd rather have neither, QED
What a paradoxical statement. “I don’t need Maths because I can think”, Obviously, you might be smart but lack intelligence. Mental calculations are included in mathematics.
@@Roq-stone 🤣 Maybe google up "affirmation of the consequent" and then take it from there, Albert. And when you use quotation marks to imply that you are quoting something, please try to get the quotation accurate, OK? Not to do so is, well, dishonest.
But with jet streams, the round trip time is reduced, not increased, because it’s not difficult to avoid the jetstream (or at least lower its velocity) as a headwind by changing altitude or route. Fly with the fastest jetstream you can, and avoid flying against a jetstream.
That visual explanation of 0.577 helps me understand statistics a little better. It pops up somewhere in the Gumbel or Pearson distribution and now I have a guess as to why.
The idea of the Wind Puzzle (A round trip always takes more time in the presence of wind) was used to disprove the existence of ether. Ether was a proposed medium that surrounds everything, and was supposed to be the medium light uses for travelling from the sun to the earth. Just like sound travels through air, ether was proposed to be a medium in which light travels. The experiment was to measure the round trip taken by light in two perpendicular directions - One in the direction of ether and the other perpendicular to ether. The light beam in one of the directions (perpendicular) was supposed to be faster, but such a difference was never recorded even though this experiment was performed in multiple orientations in multiple locations. At last, it was concluded that light is a wave that can travel in vaccuum.
It's not very well known, but since the two distances are equal, the average speed is the harmonic mean of the two velocities. And the harmonic mean is always less than or equal to the arithmetic mean, with equality if and only if the two numbers (velocities in this case) are equal. So two velocities that have an arithmetic mean of v will have a harmonic mean less than v unless both velocities were exactly v. And with a slower average velocity, the wind trip takes more time. A similar problem asks what speed you need to return to on a round trip so that your total average speed is double the first half trip's speed. (It can't be done)
Unless the boat in the mist actually stops and changes direction several times knowing the seeker will try to use this sort of predictable pattern to find it
If you ever had a swim race on a river, going both directions, you are aware of the fact that head current is slowing you down a lot more than tail current helps you.
for the first one how i got this intuitively was thinking about "how much time do i spend with time in one direction" instead of thinking about total time directly. you can somewhat see that way of thinking in the extreme case moving forward, i don't spend much time in the wind, as i already reach my destination moving backward, i get slowed down for longer. it's a classic "i move half the way at 30 and the other half at 20, what's the average speed?" question
1. even in a perfect scenario, the pilots aren't allowed to go faster than the speed of sound on commercial airlines because of the sonic boom I think so they would purposefully go slow in the fast direction then they would also go slow in the slow direction
Yeah I think they just turn up the engines when going against the wind and turn down the engines when having the wind. So the actual question would also involve how many fuel they use. Now also interesting in this scenario is that if you start with the wind against you, you will burn more fuel going, so the plane would be lighter on return. Wonder if we would take out the no wind scenario what would actually be better, starting with a headwind or returning with a headwind. I'd imagine there being a difference.
Another thing: Wind resistance increases with the cube of speed. As planes are fast, tail wind means very little. That is unless you are in a jet stream, so head wind is always several times worse. Commercial pilots usually hypermile, They fly slower in a headwind, i.e. they don't compensate completely by using the required amount of fuel to maintain speed and also throttles back in a tailwind.
You can take your first question to an even further extreme (and funnily enough, even stay within what can really happen). Get the headwind to be faster than the airspeed, and then you're going the wrong direction to cover the required distance. I've known people who have used this extreme case to land a Cesna going backwards. They didn't describe how taxiing was after landing, though
For the plane and wind problem, I would say that with the wind, the plane will stay longer at low speed than at high speed so the mean will be more affected by the low speed flight and then this case will be the slower
Thanks for very interesting videos, and putting so much effort for producing very valuable content. Even as an engineer I find them extremely interesting and easily explained of complex topics. If I can, can I suggest a video topic? (classical example of doing while asking :)) which I find personally very interesting and very few video materials available on that topic: Use of parabola and math for multiple engineering disciplines, e.g. space exploration and sound mirrors.
1:46 Fly against wind that is faster than your airplane, time becomes negative, go there before you started. I just invented the time machine. Patent pending.
For the first one, using extreme provides a really good example. Say the planes travel at 350 mph, and the wind blows at 350. From A to B is 350 miles. The plane with no wind travels the total 700 miles in two hours. The plane with wind gets a 2x speed boost and travels the first 350 miles in half the time and twice the speed of the first plane. However, on the way back, if we were to reverse the 2x speed boost the plane got on the way out, we would have to cut its speed by half, which would mean a wind speed of 175 mph. Since the wind speed is constant, it is too fast, and slows the plane down
Winds add necessary maneuvering for take off and landings if there's a tailwind. 1) The case with tailwinds on the first leg (as shown), the plane needs to take off INTO the wind so it must take off in the opposite direction of travel -- and when it lands, it must overshoot the airport and turn to land, again INTO the wind. Then on the way back, it just has headwinds; it can take off and land in the same direction of travel. 2) With headwinds on the first leg, it can just take off and land in the same direction as travel, but on the return it must oppose the direction of travel to take off and land. With no wind, no extra maneuvering, needed, thus no extra distance and time needed.
Imagine the wind speed w is greater than the velocity of the plane. Then the plane would never be able to travel from B to A, hence the time is infinite. (You can also observe this in the equation as well, if w>v, then t
There is something like the trawler problem that had real world consequences. In the 1980s a Cessna on a delivery flight to Australia lost its compass and couldn’t find its heading. A passenger plane (DC-10, IIRC) was asked to help and agreed try to find the Cessna. Passengers also agreed (and delay the flight) by looking out the windows. However to find the lost plane they first noted when and where they first made direct contact with the Cessna and then had to fly until they lost radio contact. The radio range was well known and they calculated a rendezvous point then guided the plane to safety. It is thought they flew passed the smaller plane but it was so small that it wasn’t spotted. I can’t remember the book I read this in. The Wikipedia entry has a slightly different (and probably more accurate) version. en.wikipedia.org/wiki/Cessna_188_Pacific_rescue?wprov=sfti1
For the first one an easy way to think about it is that when something is inversely related to another (like time and velocity) subtracting something from the denominator will always have a bigger affect than adding to it. This is visualized in the graph y=1/x where the slope is a lot steeper as x approaches 0.
Also, for the first problem, the opposing force seen by going opposite the wind is greater than the supportive force from the tailwind. Hence, the time is further lengthened by the presence of wind. And, the stronger the wind, the larger the difference.
for the first problem, it would be equal if the times spent under both velocities were equal. but here, you make the distances equal, resulting in more time for the deceleration to act and less time for the boost to help.
No wind is obviously faster than constant wind, but I just flew a Hawaii-Japan leg and just by going slightly south and slightly north on the way back we had an average 100 kts difference. It was glorious.
Before it starts spiraling, the chase boat is traveling twice as fast as the target, but as soon as it starts to spiral it's traveling about 9 times faster. Would have been interesting to see how big the spiral would need to be if the chase boat was "only" twice as fast (or less).
The wind example is what they were searching for, if there was an aether wind as the medium of light. The theory behind their experiment, is based on that wind example. The experiment found no results, because Galilean relativity is not applicable for speeds that are significant fractions of the speed of light, or the speed of light itself.
technically in the flatland jail example that inflates , you are always inside the fense region because if the flat person travel along the surface of the sphere at any direction he will eventually find the fence infront of him so from his perspective (since light will curve along the sphere ) the fence is still inclosing him from all directions
With regard to the first question couldn't you also use Ek=0.5mv2. The wind has certain kinetic energy that it transfers to the plane. While adding that energy will result in a lower delta v than subtracting that same amount of energy due to v squared if you consider the change as a positive number regardless of direction . Which results in more speed loss than speed gain. Hence no wind is faster than wind.
Make a video about how to enter the space industry, like whether should we get a degree in aerospace or get a degree in mechanical engineering. Ohh and whether is it possible to get a master in aerospace by having a mechanical degree?
My answer to thumnail: the same should they not be? Unless the plane is designed to catch back wind and not be affected by head wind, in which case constant wind is superior.
Gonna go through, put my guess on each out there. 0:55 top one would be faster, proof of concept (only using one example because of how extensive the typing will be). The plane flying distance D (the distance from point A to point B) with no wind, traveling at a speed without factoring in wind of S will be marked with time T and a round trip of time 2T. Using a wind blowing at a speed that’s less than the speed of the plane and assuming that there are no other forces acting on the plane you get a time that is smaller than T to travel from A to B, on the return trip however you get yourself a trip that is significantly longer than time T. For example assume the wind speed is half of that of the plane, you’d get a time between A and B of D/[(3/2)S] or 2/3 T while the trip from B to A would yield a time of D/[(1/2)S] or 2T for a total time of ~2.67T. Note there are a ton of assumptions I’m not mentioning. 3:05 I don’t know, I’d have to think about this more but my gut says not unless you can still track the wake the boat you’re chasing leaves, if you’ve got no information then you’ve got way too much ground to cover, no matter how much faster you are. Yeah, never would’ve got that but it makes total sense. 8:13 guessing this goes along the same lines of the “how many grains of sand does it take to make a pile” thing except instead it’s when is the point in which the creature is considered to no longer be inside the fence. Also entirely wrong there, I’ll take that L.
The aircraft that flies in windy conditions has to execute its landing approach and take-off into the wind at both airports. It therefore has to fly further that the aircraft flying in no wind. Flying in calm conditions, the aircraft can take-off and land in the direction of its destination on both legs. The calm flight takes less time. Sometimes, there’s more to a problem that meets the eye.
Now, here's the part that REALLY surprised me: we don't know if pi+e or pi*e is irrational, but we do know that *at least one* is irrational. So if you can prove that pi+e is *rational*, then pi*e is *irrational*. And vice versa.
@@Anda146 there's four possible combinations: both irrational, both rational, or one rational and the other irrational. Of the four, "both rational" is impossible. None of the other three have been ruled out yet
@@Anda146 e and pi are transcendental, so not the roots of any rational polynomial. (x-e)(x-pi) is a polynomial with roots e and pi; it expands to x^2-(e+pi)x+(e*pi) if both (e+pi) and (e*pi) were rational, this would be a rational polynomial, which it can't be. so if one is rational, the other has to be irrational
Coincidentally, I just did a video on this yesterday. We know that at least one of pi+e or pie is transcendental, and we also know that e^pi is transcendental. The proof above that one of them must be irrational is exactly right - f(x) = (x-e)(x-pi) is a polynomial with pi and e as roots, so its coefficients cannot all be rational. Here's a sketch to show that at least one of them is transcendental. Assume both pie and pi+e are algebraic. Use this to conclude that pi - e is also algebraic (hint: square one of them, and subtract a multiple of the other). Use this to conclude that pi is algebraic, a contradiction.
For the wind problem there's something I dont understand. Since the wind is a velocity, lets make it a vector that follows the direction of the movement. When its going in one way, the Work would be |AB| x |v| x cos(x), when its going back, the angle is now opposite so the work would be |AB| x |v| x -cos(x). Thus, the total work should equal 0, so the wind shouldnt make a difference in the airplane's speed. What's wrong in my reasoning?
Fun fact: Even though we don't know for sure if pi+e and pi*e are irrational, we know that at least one of them is. Otherwise, if pi+e and pi*e were both rational, then the solutions (namely pi and e) to the equation x^2 - (pi+e)x + pi*e = 0 would be quadratic irrationals, but we know this is not the case.
I'm not a math person but this is a cool channel! I haven't watched the video yet, only 1 minute in....But I'm going to say that the trip with wind would actually be slower round trip. I say this because there has to be a factor of time spent in such wind. The trip there, you make it faster (let's say +200mph winds), but the return trip is slower, thus more time is spent in the headwind than it was spent in the tailwind. So while you get a 200mph boost, that may last for 60 minutes, but the 200mph headwind would last longer, 80 minutes, whatever. Meaning the net result would have more time added to it since I think it would be a function of time type of thing. Again though, this is just my uneducated guess, I'm probably wrong, lol.
At 5:12, the radial component of your velocity must equal the speed of the target for this to be true. Since your velocity is mostly tangential once the spiral begins, this places a high constraint on your boat's capability vs. the target's capability. Additionally, if the target changes direction for a short time and then changes direction again, your strategy will fail because the target is always between your search paths. At 9:10, 4D space-time is hyperbolic instead of spherical; one might yet suppose, but this is unrealistic.
For the jet I would guess it depends on the design. Some have a design that minimizes the amount of resistance flying in to head wind but have a somewhat greater force from a tail wind
The premise is that the wind has a uniform velocity profile, so that the aircraft cannot tell the difference between a headwind and a tailwind. The airplane is assumed to have the same speed in both winds, and only have that speed relative to the wind. A real wind will have a non-uniform velocity profile, which is what an airplane take advantage of for being better designed to fly with a headwind.
Move a little and the small ballon on the other side of the universe no longer fills the horizon, and things look more sensible. Better still to start near the edge of it rather than in the middle.
The still air vs constant wind journey scenario can be asnswered much more simply using common sense. With a wind, any tailwind acts to improve groundspeed for less time and the headwing acts to reduce groundspeed for longer.
That still isn't enough to tell the difference between a still wind and a round trip with and against a constant/uniform headwind/tailwind. You need to set up the problem mathematically to tell the difference. A general common sense statement that tailwinds improve ground speed and headwinds reduce ground speed, isn't enough reasoning to conclude that it is any different from the case with no wind.
@@axelnils Apologies for taking so long to reply to this. I actually found that Zach was "right" but you have to make assumptions that he himself did not state he was making. Let me start by using the assumption that I imagine most would use: no thrust force is applied to the plane. With this assumption in play, the planes would never move at a constant velocity as even in the no wind scenario, air resistance is still a thing. In fact, even with a tailwind, the second plane would move slower than in a vacuum until the wind matched or was faster than the plane's own speed. Furthermore, there is not a chance that the second plane's velocity would be impacted that much by w (assuming w is the wind speed). Imagine you are jumping around in gale force winds for science or whatever. You will certainly be affected by the winds but not nearly enough to fling you around at twice the speed of Usain Bolt. Conversely, a 1 m/s wind is not going to do anything to you walking at 1 m/s. In fact, you will barely feel it. To work out this problem, you would have to deal with some differential equations that I'm just far to lazy to solve. All that being said, Zach can be completely right if a) Thrust forces are applied such that the velocities are as Zach said. b) The drag is directly proportional to the the plane's relative speed to the air. (The standard model of using the square of the plane's relative speed also works for Zach but the logic differs a bit). Assumptions a and b allow for the planes to have the same resultant thrust force applied. Of course, you could allow the planes to have different thrusts but that kind of defeats the purpose of the problem.
I know, random, but i really gotta get a refresher. @Zach Star or anyone: Which video was it, where he talked about likely choosing the best (canidate) for a job or whatever, (something about forget the first 3rd of the elements of the set, then sift through the ones from 1/3 into the pile and pick the first great option you see) im not sure what the actual context was for the example, but yeah, that whole idea
My intuition about the wind problem was that drag forces are cubic with velocity, so gains going with the wind are not compensated for by gains going against it.
I took the airplane one even more extreme. If the wind is faster than the plane, you will never get back to the starting point. So if the wind's effect on travel time is monotonic (which this thought sample does not prove, for the record), it will increase travel time.
The wind senario doesn't make sense though, headwind gives a plane lift which can make said plane faster! And there is no limit set on the question to how complex the answer can be which means the answer can be either faster or slower.
The last one is either mind boggling or trivial. Depending on how one looks at it. Wouldn't the balloon (if we move to the 3D example) turn inside out so the inner skin would be outside when it "flips"?
Waking up to watching this video in the morning in India and then for the whole day thinking about the times taken by plane, Euler Mascheroni constant, irrationality/rationality of pi+e, the Trawler problem and how to theoretically escape from that balloon etc. The day is fun!
Hope you guys enjoy! As stated at the end I'll taking a little holiday break so will return in the New Year (possibly with a new skit video). You can check out the second channel ruclips.net/user/zachstarhimself for exclusively skit videos and as always check out STEMerch.com to get the floating globe, meme shirts, and more.
Have a nice holiday!
Hey can you do a video on the dream Minecraft speed running statistics and why he got banned
Hey Zach, can you plz do a video on mechatronics engineering. Please I would really appreciate that thanks.
The fog boat is just the chase of the Bismarck
That was awesome but the first one is utterly counterintuitive and I think needs a video of its own...really...shouldn't it travel fast enough in the forward direction due to the same velocity of wind taking it in the first direction to make up for the slowing caused in the second direction...the first direction might almost be instantaneous whereas second one takes the amount of time taken by the no wind plane to go back and forth?...please explain...thanks a lot.
Everybody gangsta until the cop car chase goes into a logarithmic spiral...
lmao
Doesn't really work with cars unless it's in a city with logarithmic spiral streets :,D
just change direction 2 times
Or just drive slower at a different velocity.
Or just don't commit crimes :D
How does a mathematician capture all lions in Africa? He builds a cage, steps into the cage and defines: "This is outside."
That's hilarious.
He’s thinking outside the box
And then he gets eaten by a lion that was outside the cage.
@Butch (the black cat from tom and Jerry) He missed that a lion had snuck into the cage
@@Etazoz *out of
*Physics strategy of taking things to the extreme*
"Now, assuming this cow is perfect sphere,"
@@AmmoGus1 and the sun doesnt exist too, so photons wont push it in any direction and no energy is being added to the system
And pretend the cosmic microwave background doesn't exist, and neither does heat transfer by radiation.
Oh yes, and all collisions are perfectly elastic
don't forget to Assume all gasses are ideal and the system is isolated
-"Captain we are approaching heavy fog"
An appropriate amount of time later
-"Adopt a logarithmic spiral trajectory"
Also accelerate to faster than light to continue the search fast enuff!
Faster method for the 'fog-boat' thing: start the spiral when you would have encountered the boat if it turned 180*
Which boat as shown in the animation of the video would u go for 1st tho?
@@bamb8s436 the one which goes straight to the left. It was for some reason omitted from the visualization.
David.
Interesting thought!
Why do you think this? It's certainly an attractive idea because you start your spiraling earlier. but the spiraling isn't the essence of what you're about, surely. You want to find the other boat earlier and since 50% 0f the time the spiral is away from the target, no matter which way it chose to flee, therefore 50% of the time your early spiral is giving the target extra time in which to get more distant, surely?
A thought which may or may not be equivalent, I'm not sure, in 50% of all cases starting early to veer away from your path straight ahead on known information is adding to the time the target has to add to its distance from the last place you saw it, don't you think?
@@lordfnord5768 if one thinks in worst cases, my method is better. Once you start spiraling, you have to max do one round. Since I both use less distance to where I start and have a smaller spiral (the chased boat haven't gotten as far when I reach it) it will be faster.
@@lordfnord5768 also, in the case in the video the 50% who turn backwards make that spiral longer. So it is the same either case, but i travel less before starting spiraling.
Often this reminds that our 4-5 grade mathsbooks contained more word problems involving real life question , but in 9 grade chapters word problems involving real problems are as rare as Halley's comet
I see you are a man of culture as well
mathsbooks are the only place where someone can buy 54 watermelons and no one asks why
@@howiestillgamez5326 Real men buy log(43) melons
the way I saw the first one is you'll be slowed down for more time than you'll be sped up, so the slowing down is more effective and it'll be slower on the whole.
Joke of the day:
Did you hear about the claustrophobic astronaut?
*He needs a lot of space.*
*pokerface*
Get your hands off my face bro
ruclips.net/video/KZFcYK2VzkI/видео.html
YOOO brother
thanks to you i didn't catch the boat but just crashed into it.
You may be "outside" the balloon, but you would still be separated from the rest of the universe, which would now be "inside" the balloon.
Have you seen the comic called “The Fence” by Daniel Quinn? It’s a good illustration of this on a 2-sphere.
@@ragnkja No, I haven't. Interesting.
6:55 : I have discovered a truly remarkable proof of the irrationality of pi plus e, but the margin of this RUclips comment is too small to contain Fermat's last theorem.
Go up 🔝🚀
Lmao, nice historic math reference!
I thought he said it hasn't been proven whether or not it's irrational?
@@anshumanagrawal346 It's a joke referencing Fermat's Last Theorem.
See: en.wikipedia.org/wiki/Fermat's_Last_Theorem
2:07 lol that was my first thought when I started the problem, just “what if you can’t move going back from the wind being too fast”
My intuition for the first puzzle is similar. If we consider a headwind with equal velocity to the plane's, clearly the return trip takes an infinite amount of time. Therefore, it can't always be equal.
No shit.
Me a physics boi: wouldn’t the limiting case be the plane takes infinite time when W=V.
Zach 0.2 nano second after: 2:15
YO im also a physics boi
Edit for clarity: 200 ps
Suppose wind flowing in direction a to b now at first plane will reach twice fast but when returning as wind speed against plane is same , the plane velocity will be 0 and plane won't move which ultimately makes time infinite 🥱.
If W >V, the divisor becomes negative and so the time becomes negative. I just invented a time machine.
Wouldn't be the first time I escaped from a balloon when I wasn't supposed to.
damn
damn
damn
This was really interesting. Merry Christmas!
"I don't need any math, I can think." -- Joan Robinson, English economist.
The wind is going to slow you down one way. Going slower you will be slowed down for a long time.
The wind is going to speed you up going the other way. Speeded up you are going to be speeded up for a short time.
You're speeded up by the same speed as you're slowed down by -- but there's more slowing down than speeding up going on, so you'd rather have neither, QED
What a paradoxical statement.
“I don’t need Maths because I can think”,
Obviously, you might be smart but lack intelligence.
Mental calculations are included in mathematics.
So please find the area under the graph of x^x from 0 to 1 using language
@@gammarayneutrino8413 that seems like an unfair test, since you didn't state the problem using natural language
Savethiscomment
@@Roq-stone
🤣
Maybe google up "affirmation of the consequent" and then take it from there, Albert.
And when you use quotation marks to imply that you are quoting something, please try to get the quotation accurate, OK? Not to do so is, well, dishonest.
Reminds me of jet streams which jets going east often use to save fuel
But with jet streams, the round trip time is reduced, not increased, because it’s not difficult to avoid the jetstream (or at least lower its velocity) as a headwind by changing altitude or route. Fly with the fastest jetstream you can, and avoid flying against a jetstream.
@@geoffstrickler Yes pilots often avoid the jet streams when going against them as they are narrow tubes of current
That boat scenario was super relatable, thank you I no longer feel alone
Interesting how things turn out to go when you consider many factors!
That visual explanation of 0.577 helps me understand statistics a little better. It pops up somewhere in the Gumbel or Pearson distribution and now I have a guess as to why.
As a student of Piratical Engineering the second example was super relevant to me
The idea of the Wind Puzzle (A round trip always takes more time in the presence of wind) was used to disprove the existence of ether.
Ether was a proposed medium that surrounds everything, and was supposed to be the medium light uses for travelling from the sun to the earth. Just like sound travels through air, ether was proposed to be a medium in which light travels. The experiment was to measure the round trip taken by light in two perpendicular directions - One in the direction of ether and the other perpendicular to ether. The light beam in one of the directions (perpendicular) was supposed to be faster, but such a difference was never recorded even though this experiment was performed in multiple orientations in multiple locations.
At last, it was concluded that light is a wave that can travel in vaccuum.
I don’t want it to be very useful, I just like the original educational content
You know, we can just HEAR the other boat
This guy sounds just like those two guys that kidnapped the wrong girl
Wut
Go watch his second channel
I just wanna get a reply from Zach. I have never gotten a reply from him. It's my last chance this year.
Here you go!
@@zachstar omg
Lmao what
@@ViratKohli-jj3wj What a mad lad
You should follow up by asking to be an extra in an upcoming video or something.
It's not very well known, but since the two distances are equal, the average speed is the harmonic mean of the two velocities. And the harmonic mean is always less than or equal to the arithmetic mean, with equality if and only if the two numbers (velocities in this case) are equal.
So two velocities that have an arithmetic mean of v will have a harmonic mean less than v unless both velocities were exactly v. And with a slower average velocity, the wind trip takes more time.
A similar problem asks what speed you need to return to on a round trip so that your total average speed is double the first half trip's speed. (It can't be done)
Unless the boat in the mist actually stops and changes direction several times knowing the seeker will try to use this sort of predictable pattern to find it
If you ever had a swim race on a river, going both directions, you are aware of the fact that head current is slowing you down a lot more than tail current helps you.
the oily-macaroni constant lmaoo
for the first one
how i got this intuitively was thinking about "how much time do i spend with time in one direction" instead of thinking about total time directly. you can somewhat see that way of thinking in the extreme case
moving forward, i don't spend much time in the wind, as i already reach my destination
moving backward, i get slowed down for longer.
it's a classic "i move half the way at 30 and the other half at 20, what's the average speed?" question
1. even in a perfect scenario, the pilots aren't allowed to go faster than the speed of sound on commercial airlines because of the sonic boom I think so they would purposefully go slow in the fast direction then they would also go slow in the slow direction
Yeah I think they just turn up the engines when going against the wind and turn down the engines when having the wind. So the actual question would also involve how many fuel they use. Now also interesting in this scenario is that if you start with the wind against you, you will burn more fuel going, so the plane would be lighter on return. Wonder if we would take out the no wind scenario what would actually be better, starting with a headwind or returning with a headwind. I'd imagine there being a difference.
Another thing: Wind resistance increases with the cube of speed. As planes are fast, tail wind means very little. That is unless you are in a jet stream, so head wind is always several times worse. Commercial pilots usually hypermile, They fly slower in a headwind, i.e. they don't compensate completely by using the required amount of fuel to maintain speed and also throttles back in a tailwind.
You can take your first question to an even further extreme (and funnily enough, even stay within what can really happen). Get the headwind to be faster than the airspeed, and then you're going the wrong direction to cover the required distance. I've known people who have used this extreme case to land a Cesna going backwards. They didn't describe how taxiing was after landing, though
For the plane and wind problem, I would say that with the wind, the plane will stay longer at low speed than at high speed so the mean will be more affected by the low speed flight and then this case will be the slower
Thanks for very interesting videos, and putting so much effort for producing very valuable content. Even as an engineer I find them extremely interesting and easily explained of complex topics. If I can, can I suggest a video topic? (classical example of doing while asking :)) which I find personally very interesting and very few video materials available on that topic: Use of parabola and math for multiple engineering disciplines, e.g. space exploration and sound mirrors.
The Euler-Mascheroni constant looks suspiciously close to 1/ sqrt(3).
random things for a random year
this needs more upvotes
"For this next one it will be completely relatable"
"Big boat goes invisible"
Zach be making me laugh and learn in the same day. Great job here and on the other channel too Zach!
You can prove the answer to the first question using the harmonic mean.
1:46 Fly against wind that is faster than your airplane, time becomes negative, go there before you started. I just invented the time machine. Patent pending.
Hopefully next year is better.
For the first one, using extreme provides a really good example. Say the planes travel at 350 mph, and the wind blows at 350. From A to B is 350 miles. The plane with no wind travels the total 700 miles in two hours. The plane with wind gets a 2x speed boost and travels the first 350 miles in half the time and twice the speed of the first plane. However, on the way back, if we were to reverse the 2x speed boost the plane got on the way out, we would have to cut its speed by half, which would mean a wind speed of 175 mph. Since the wind speed is constant, it is too fast, and slows the plane down
Winds add necessary maneuvering for take off and landings if there's a tailwind.
1) The case with tailwinds on the first leg (as shown), the plane needs to take off INTO the wind so it must take off in the opposite direction of travel -- and when it lands, it must overshoot the airport and turn to land, again INTO the wind. Then on the way back, it just has headwinds; it can take off and land in the same direction of travel.
2) With headwinds on the first leg, it can just take off and land in the same direction as travel, but on the return it must oppose the direction of travel to take off and land.
With no wind, no extra maneuvering, needed, thus no extra distance and time needed.
All of that is assumed to take a negligible amount of time, compared to the essence of the problem.
Imagine the wind speed w is greater than the velocity of the plane. Then the plane would never be able to travel from B to A, hence the time is infinite. (You can also observe this in the equation as well, if w>v, then t
2nd problem is very applicable in real life especially in land Nav. Although squaring away is a much preferred strategy, being land Nav and all.
There is something like the trawler problem that had real world consequences.
In the 1980s a Cessna on a delivery flight to Australia lost its compass and couldn’t find its heading.
A passenger plane (DC-10, IIRC) was asked to help and agreed try to find the Cessna. Passengers also agreed (and delay the flight) by looking out the windows.
However to find the lost plane they first noted when and where they first made direct contact with the Cessna and then had to fly until they lost radio contact. The radio range was well known and they calculated a rendezvous point then guided the plane to safety.
It is thought they flew passed the smaller plane but it was so small that it wasn’t spotted.
I can’t remember the book I read this in.
The Wikipedia entry has a slightly different (and probably more accurate) version.
en.wikipedia.org/wiki/Cessna_188_Pacific_rescue?wprov=sfti1
For the first one an easy way to think about it is that when something is inversely related to another (like time and velocity) subtracting something from the denominator will always have a bigger affect than adding to it. This is visualized in the graph y=1/x where the slope is a lot steeper as x approaches 0.
Also, for the first problem, the opposing force seen by going opposite the wind is greater than the supportive force from the tailwind. Hence, the time is further lengthened by the presence of wind. And, the stronger the wind, the larger the difference.
for the first problem, it would be equal if the times spent under both velocities were equal.
but here, you make the distances equal, resulting in more time for the deceleration to act and less time for the boost to help.
I'd love to see more unsolved maths problems in future videos
I was binge watching your skits. I genuinely watched 8 minutes of this before i was like, wait, this one isnt a a joke.
No wind is obviously faster than constant wind, but I just flew a Hawaii-Japan leg and just by going slightly south and slightly north on the way back we had an average 100 kts difference. It was glorious.
The plane problem is similar to the doppler effect with a moving source.
Before it starts spiraling, the chase boat is traveling twice as fast as the target, but as soon as it starts to spiral it's traveling about 9 times faster. Would have been interesting to see how big the spiral would need to be if the chase boat was "only" twice as fast (or less).
The wind example is effectively why the Michelson and Morley experiment could show no aether wind for the speed of light.
The wind example is what they were searching for, if there was an aether wind as the medium of light. The theory behind their experiment, is based on that wind example. The experiment found no results, because Galilean relativity is not applicable for speeds that are significant fractions of the speed of light, or the speed of light itself.
technically in the flatland jail example that inflates , you are always inside the fense region because if the flat person travel along the surface of the sphere at any direction he will eventually find the fence infront of him so from his perspective (since light will curve along the sphere ) the fence is still inclosing him from all directions
With regard to the first question couldn't you also use Ek=0.5mv2. The wind has certain kinetic energy that it transfers to the plane. While adding that energy will result in a lower delta v than subtracting that same amount of energy due to v squared if you consider the change as a positive number regardless of direction . Which results in more speed loss than speed gain. Hence no wind is faster than wind.
Cranks on that imaginary wind to get a faster flight than I would with no wind.
Make a video about how to enter the space industry, like whether should we get a degree in aerospace or get a degree in mechanical engineering. Ohh and whether is it possible to get a master in aerospace by having a mechanical degree?
My answer to thumnail: the same should they not be? Unless the plane is designed to catch back wind and not be affected by head wind, in which case constant wind is superior.
Gonna go through, put my guess on each out there.
0:55 top one would be faster, proof of concept (only using one example because of how extensive the typing will be). The plane flying distance D (the distance from point A to point B) with no wind, traveling at a speed without factoring in wind of S will be marked with time T and a round trip of time 2T. Using a wind blowing at a speed that’s less than the speed of the plane and assuming that there are no other forces acting on the plane you get a time that is smaller than T to travel from A to B, on the return trip however you get yourself a trip that is significantly longer than time T. For example assume the wind speed is half of that of the plane, you’d get a time between A and B of D/[(3/2)S] or 2/3 T while the trip from B to A would yield a time of D/[(1/2)S] or 2T for a total time of ~2.67T. Note there are a ton of assumptions I’m not mentioning.
3:05 I don’t know, I’d have to think about this more but my gut says not unless you can still track the wake the boat you’re chasing leaves, if you’ve got no information then you’ve got way too much ground to cover, no matter how much faster you are. Yeah, never would’ve got that but it makes total sense.
8:13 guessing this goes along the same lines of the “how many grains of sand does it take to make a pile” thing except instead it’s when is the point in which the creature is considered to no longer be inside the fence. Also entirely wrong there, I’ll take that L.
The aircraft that flies in windy conditions has to execute its landing approach and take-off into the wind at both airports. It therefore has to fly further that the aircraft flying in no wind. Flying in calm conditions, the aircraft can take-off and land in the direction of its destination on both legs.
The calm flight takes less time.
Sometimes, there’s more to a problem that meets the eye.
Now, here's the part that REALLY surprised me: we don't know if pi+e or pi*e is irrational, but we do know that *at least one* is irrational. So if you can prove that pi+e is *rational*, then pi*e is *irrational*. And vice versa.
Yea source please. Im interested.
Although simply asuming is dangerous I still believe that both are irrational
pi and e are transcendental, so they can't both be solutions to a quadratic equation. Therefore at least one of pi+e and pi*e is irrational.
@@Anda146 there's four possible combinations: both irrational, both rational, or one rational and the other irrational. Of the four, "both rational" is impossible. None of the other three have been ruled out yet
@@Anda146 e and pi are transcendental, so not the roots of any rational polynomial. (x-e)(x-pi) is a polynomial with roots e and pi; it expands to x^2-(e+pi)x+(e*pi)
if both (e+pi) and (e*pi) were rational, this would be a rational polynomial, which it can't be. so if one is rational, the other has to be irrational
Coincidentally, I just did a video on this yesterday. We know that at least one of pi+e or pie is transcendental, and we also know that e^pi is transcendental.
The proof above that one of them must be irrational is exactly right - f(x) = (x-e)(x-pi) is a polynomial with pi and e as roots, so its coefficients cannot all be rational.
Here's a sketch to show that at least one of them is transcendental. Assume both pie and pi+e are algebraic. Use this to conclude that pi - e is also algebraic (hint: square one of them, and subtract a multiple of the other). Use this to conclude that pi is algebraic, a contradiction.
"So long as the chasing boat is going significantly faster than the other." interesting...
Thanks, good holiday!
Flying through headwind is actually faster than with tailwind since it creates more lift.
For the wind problem there's something I dont understand. Since the wind is a velocity, lets make it a vector that follows the direction of the movement. When its going in one way, the Work would be |AB| x |v| x cos(x), when its going back, the angle is now opposite so the work would be |AB| x |v| x -cos(x). Thus, the total work should equal 0, so the wind shouldnt make a difference in the airplane's speed. What's wrong in my reasoning?
Fun fact: Even though we don't know for sure if pi+e and pi*e are irrational, we know that at least one of them is. Otherwise, if pi+e and pi*e were both rational, then the solutions (namely pi and e) to the equation x^2 - (pi+e)x + pi*e = 0 would be quadratic irrationals, but we know this is not the case.
These are my favourites, thanks for this video❤❤
Worth mentioning is that it is known that at least one of π + e and π x e is irrational.
This comes directly from the fact that pi and e are both transcendental. Actually only one of them has to be transcendental.
4:12 Animator: So you're telling me my copy-paste skills are *not* mathematically precise? Find yourself a new animator! 😂
Simple reckoning, flying into wind time will be longer so negative effect of wind is greater than positive effect on ‘with wind’ leg.
I'm not a math person but this is a cool channel! I haven't watched the video yet, only 1 minute in....But I'm going to say that the trip with wind would actually be slower round trip. I say this because there has to be a factor of time spent in such wind. The trip there, you make it faster (let's say +200mph winds), but the return trip is slower, thus more time is spent in the headwind than it was spent in the tailwind. So while you get a 200mph boost, that may last for 60 minutes, but the 200mph headwind would last longer, 80 minutes, whatever. Meaning the net result would have more time added to it since I think it would be a function of time type of thing. Again though, this is just my uneducated guess, I'm probably wrong, lol.
At 5:12, the radial component of your velocity must equal the speed of the target for this to be true. Since your velocity is mostly tangential once the spiral begins, this places a high constraint on your boat's capability vs. the target's capability. Additionally, if the target changes direction for a short time and then changes direction again, your strategy will fail because the target is always between your search paths. At 9:10, 4D space-time is hyperbolic instead of spherical; one might yet suppose, but this is unrealistic.
baloon 4d sphere thing means that every point in closed universe would have it's antipode. Never thought about that
“How does this first one affect my life”
Me, a CFI: “ummmmmm”
For the jet I would guess it depends on the design. Some have a design that minimizes the amount of resistance flying in to head wind but have a somewhat greater force from a tail wind
The premise is that the wind has a uniform velocity profile, so that the aircraft cannot tell the difference between a headwind and a tailwind. The airplane is assumed to have the same speed in both winds, and only have that speed relative to the wind.
A real wind will have a non-uniform velocity profile, which is what an airplane take advantage of for being better designed to fly with a headwind.
Move a little and the small ballon on the other side of the universe no longer fills the horizon, and things look more sensible. Better still to start near the edge of it rather than in the middle.
The still air vs constant wind journey scenario can be asnswered much more simply using common sense. With a wind, any tailwind acts to improve groundspeed for less time and the headwing acts to reduce groundspeed for longer.
That still isn't enough to tell the difference between a still wind and a round trip with and against a constant/uniform headwind/tailwind. You need to set up the problem mathematically to tell the difference. A general common sense statement that tailwinds improve ground speed and headwinds reduce ground speed, isn't enough reasoning to conclude that it is any different from the case with no wind.
I think you've greatly oversimplified the wind problem here, Zach.
How?
@@axelnils Apologies for taking so long to reply to this. I actually found that Zach was "right" but you have to make assumptions that he himself did not state he was making. Let me start by using the assumption that I imagine most would use: no thrust force is applied to the plane. With this assumption in play, the planes would never move at a constant velocity as even in the no wind scenario, air resistance is still a thing. In fact, even with a tailwind, the second plane would move slower than in a vacuum until the wind matched or was faster than the plane's own speed. Furthermore, there is not a chance that the second plane's velocity would be impacted that much by w (assuming w is the wind speed). Imagine you are jumping around in gale force winds for science or whatever. You will certainly be affected by the winds but not nearly enough to fling you around at twice the speed of Usain Bolt. Conversely, a 1 m/s wind is not going to do anything to you walking at 1 m/s. In fact, you will barely feel it. To work out this problem, you would have to deal with some differential equations that I'm just far to lazy to solve.
All that being said, Zach can be completely right if
a) Thrust forces are applied such that the velocities are as Zach said.
b) The drag is directly proportional to the the plane's relative speed to the air. (The standard model of using the square of the plane's relative speed also works for Zach but the logic differs a bit).
Assumptions a and b allow for the planes to have the same resultant thrust force applied. Of course, you could allow the planes to have different thrusts but that kind of defeats the purpose of the problem.
The trawler one was very cool
I know, random, but i really gotta get a refresher.
@Zach Star or anyone:
Which video was it, where he talked about likely choosing the best (canidate) for a job or whatever, (something about forget the first 3rd of the elements of the set, then sift through the ones from 1/3 into the pile and pick the first great option you see) im not sure what the actual context was for the example, but yeah, that whole idea
My intuition about the wind problem was that drag forces are cubic with velocity, so gains going with the wind are not compensated for by gains going against it.
Drag is irrelevant. Groundspeed is everything. The headwind acts on the aircraft for longer than the tailwind.
The headwind slowing you down means you spend more time in headwind.
Additionally,
1+1
Definitely gonna escape that infinitely strachey balloon. THX for that...
I really enjoy your content!!
I can feel smart while not understanding anything
1. U can think of it like that: you fly with boost for less time than with anti boost
The plane result is counterintuitive.
I took the airplane one even more extreme. If the wind is faster than the plane, you will never get back to the starting point. So if the wind's effect on travel time is monotonic (which this thought sample does not prove, for the record), it will increase travel time.
The wind senario doesn't make sense though, headwind gives a plane lift which can make said plane faster! And there is no limit set on the question to how complex the answer can be which means the answer can be either faster or slower.
This is why i just love physics
Yeah the top plane wins the bottom one falls down.
Imagine wind speed approaching speed of plane in 2nd case
It's out of my *limits*
For someone : w²/v approaches v (from LH) v-w²/v approaches 0, 1/(v-w²/v) approaches ______
π+e=6
π-e=0
πe=9
π^e=27
Here you go, in your face Euler- mascheroni constant
More like I massacred your only constant😂
Well if
π=e
And
π^2=10
=> π=√10
e=√10
Thus,
π+e=2√10. So
6=2×√10
The last one is either mind boggling or trivial. Depending on how one looks at it. Wouldn't the balloon (if we move to the 3D example) turn inside out so the inner skin would be outside when it "flips"?
Waking up to watching this video in the morning in India and then for the whole day thinking about the times taken by plane, Euler Mascheroni constant, irrationality/rationality of pi+e, the Trawler problem and how to theoretically escape from that balloon etc. The day is fun!