@@hisanuswat4359i've heard people get shit like this instead: "There are 2 trains about to collapse into each other at X speed, how fast does it takes for a fly in X speed to touch both of them before they clash ?" Not entirely correct to the actual question, but something similar.
@@dev4159 I guess it's similar. I mean, I myself have never found exactly this problem on tests, but similar ones, but we usually got such problems in vectors and 2 dimensional motion stuff
Next time you make a video on how to not get hit by a car, could you make it a bit shorter? The car is getting closer and I still have several minutes left in the video
Better question: If, hypothetically, there was a truck trying to hit a person, what is the path they must take to most optimally try to hit them given that the person is taking the most optimal path to run away?
How about: If, hypothetically, there was a truck trying not to hit a person, what is the path they must take to not run into the path the truck is swerving to not hit them?
Go *straight* to jail. Do not pass go! You are asking for classified and export controlled information. Please turn yourself in to the nearest USAF base immediately. No it doesn't matter that you only asked about a 2D representation of terminal guidance that has been public knowledge for more than half a century.
When I was in high school, we started learning about polar functions, including limaçons. Separately, I noticed that when an overhead light shined on a mug of milk, the light made a pattern on the milk that looked suspiciously like a limaçon. I was able to prove it, and I challenge Math The World to do the same!
Yes lol, it is very annoying when someone uses "feet" "miles" "Fahrenheit" or something... I have to convert them to metric system just to understand them
I don't see why you feel the need to convert the units to metric. The units don't matter. It works exactly the same no matter what units you use. He might as well have used furlongs per fortnight or potrzebies per clarke. Lightyears per plank time, maybe.
Also, when the units do matter we feel the same way about someone using meters, °C, etc. We have to convert to feet, °F, etc. to understand. Why would I make my problem not understandable to myself to make it understandable to some random person on the internet? That's what truly doesn't make sense.
@@WhirlwindQuestEver seen moonwalking? Spoiler alert: you do not need to turn around to walk back. Honestly, jumping backwards is probably the fastest dodge.
It makes sense that it would be based on the ratio of speeds, since changing both speeds by the same factor is equivalent to running an identical scenario in either slow motion or fast forward.
That’s a fair point, but I would add that it’s still somewhat enlightening that the solution turns out to be ONLY dependent on that ratio. My gut assumption was that the ratio of speeds would be the biggest contributor, but the distance from the curb and the distance from the truck would also matter. (They do, but only in the binary sense of “will you make it or not?”)
@@pjl22222 1 m/s = 3.6 km/h and the ratio 3.6 comes from 3600/1000 (number of seconds in 1h)/(number of meters in 1km) so it's not a power of 10 but 3.6
@@pjl22222 it’s easier to look at it this way: imagine you have to convert x km/h to m/s. x km/h = x km/h*1*1= x km/h * 1000 m/km * 1h/3600s Now the km and h cancel out leaving you with x*1000/3600 m/s = x*10/36 m/s = x*5/18 m/s
Cool video, but the fastest way to go is backwards. Made even faster because trying to turn takes time, while when going backwards you don’t need to turn at all.
Actually, it is a good question about when is it quicker to go backwards and when is it better to go forward. You already have momentum going forward, so presumably, in the same time period, you could go further going forward than backwards, which would require turning or stopping/walking backwards. There is also the issue with the traffic in the lane behind you that you also have to avoid.
@@ExzaktVid Many years back, I had an accident cycling through an intersection with a car crossing. Minimal background info: I have no 3d sight, so am bad at guessing distances and speeds. Experience helps, so long as situations are consistent with speeds - turns out (according to other people), the car was speeding. The problem I had was, essentially, indecisiveness - I changed several times my opinion if I should try to beat the car, or if it would be better to stop. turns out the result was a collision where I was pretty much unhurt and the bike pretty much undamaged - no real idea about the car, because by the time it managed to stop, it was a good bit beyond the intersection and I couldn't see the sides. The driver obviously thought they were not free of guilt, from them only ever asking if I was ok. Obviously, what would have been right depends on speeds and positions ... which I wasn't best at guessing, but the math might still be interesting.
Again, the backward strategy is just assuming you don't already have strong forward momentum, otherwise you would just slide across the concret (and likely fall too)
in theory the Truck should be trying to slow down (that's also what saved that kid in the opening video) Could try to explore what would be happening with a truck speed reducing, but not enough to totally be at rest before meeting the perpendicular. Nice video otherwise. That's a very good example of Calculus power !
Yes we definitely took a worse case scenario here of if the truck never saw you and continues at its speed! But if it slowed down then that's even better for Sara!
@@MathTheWorldit's even better for Sara, but that may mean there's an even larger benefit for moving away from the truck, because the truck would be going slower the more you increase the distance away from the truck. It may make the optimal angle steeper, but I do think the math gets much harder at that point
Yeah, when this situation happens, I'd advise Sara to inquire about the breaking capabilities of the specific type of truck from the manufacturer first, and to also weight in the degradation of the specific truck's breaks to come up with the optimal solution. X) But to be real, the angles are likely a lot different when considering breaking, especially if we consider that it doesn't matter too much which part of the truck hits us (as long as the front is flat), but it matter a lot, how fast it hits us. So if you just cannot avoid being hit, then it may be worth to run straight away, so the truck causes the least damage to you (not only due to breaking, but also because your speed is also subtracted from the truck's "damaging potential" when the hitting happens)
@@MathTheWorldto do this you could use integration. If the velocity of the truck is defined by v0-x (v0=initial velocity), its integral, and therefore its position, would be defined by s0+v0*x-(x^2)/2. I believe this would greatly change the angle. Im not going to follow through on the math right now, but I might some other day!
@@MathTheWorld Which is good because if the truck sees you the analysis gets really complicated with how the driver responds to seeing you and how that affects the optimal route.
Favorite math problem.🤔 The is tuff. I love linear algebra. But the thing i use the most to make sense of the world are simple energy analisys, simple heat flow and unit analysis. Combined with +-*/ and % calculations you can get a solid understanding or approximation of almost any day to day problem.
@@janthran I have only done very simple steady state circuit analysis many years ago, so I don't know how alike they are. But they at least can have the steady state assumption in common.🤓
The answer is backwards, out of the way of the truck and waiting until it is past. Boom! No math needed lol. 😄. This is a really interesting video though. I apologize if I came across as argumentative, I was trying to be humorous, but I'm autistic and sometimes things don't come across as I intend them.
Yeah, we worried viewers might think that with the position we drew the person. We should have drawn another vehicle coming up the other lane, so then that option is even more dangerous than going forward. It certainly it is a valid option. I wonder at what point is it better to turn around/stop and go backwards vs going forward, since you already have momentum going forward. I don't know if a mathematical model would give you a very good estimate here, but it is an interesting question.
@@pjl22222 yes, turning around is what causes squirrels to get flattened. But humans have the ability to move backwards without having stop and turn around. It's one of the benefits of being bipedal.
This is THE BEST VIDEO explaining what are the the differences between different math levels and WHY we invented calculus. THANK YOU SO MUCH. I will share this video a lot
The bigger the truck is, the harder it is to estimate its speed, the bigger the uncertainty of that factor. If you run straight away from the truck, then you also minimize speed difference. Let's say your running speed is 10 km/h and the truck comes incredibly fast and breaks incredibly fast, so that you have 0 chance to avoid being hit and your angle won't significantly change the time when you get hit or the speed of the truck at the moment of collision, but because of incredible strong breaks the trucks speed at that point in time is just 15 km/h. So if you run straight away from it, then you get hit with a relative speed of just 5 km/h, which you are very likely to survive. If the trucker releases the breaks as soon as he reaches 5 km/h, then you'll get run over slowly and still die, but let's not assume that. If you run perpendicular, then the relative speed is sqr(10²+15²)≈18km/h which is almost impossible to survive. In fact if the truck driver didn't buckle his seat belt and you're a giant block of concrete, then there's a pretty good chance that the truck driver dies. Ironically in this case (minus the concrete block thing) your chances of survival would be marginally bigger if you just stood still. Don't do that though.
My solution: dont cross the street without taking a good look and take a break for a few seconds and then cross. Also dont cross between other cars when you cant see the other lane. Again take a break and look before crossing
Yes, this is the best solution. Your point about crossing when you can't see both lanes (or all 4 lanes on a larger street) is so important. We had a couple of kids killed here in Utah a few years ago because of a well-meaning driver stopping on a 4 lane road to let the kids cross in front of his/her car. Well, his/her big SUV blocked the view of other drivers in the lane next to him/her, and couldn't see the kids, and the kids couldn't see the car in the other lane either. As soon as they cleared the SUV and got into the second lane, they were immediately hit by a car. It was so sad.
Why are so many people complaining about the units 💀 I’m a metric guy, but the units used have absolutely 0 bearing on the underlying math or visualizations. He could’ve even not used any system and just said “units per time interval” and it’d all still make perfect sense
Great video. I think one thing that might help the viewer look for insights would be to highlight that even before you pointed it out at 9:05, we already knew that it doesn't matter what the actual speeds are, because we know that the actual distances don't matter. If it doesn't matter that the truck is twice as far away, or that the curb is closer or further, then it stands to reason that the actual speed doesn't matter.
It's not clear to me that maximizing distance between you and the truck at the curb is the right thing to optimize, as opposed to say, minimizing the time to the curb among all paths which keep you safe. And as you noticed, this model behaves weirdly when r is at least 1. Sure, it's trivial to survive when r > 1, but if r = 1, this model prescribes running parallel to the curb forever, and fails to ascertain whether there is any safe way to reach the curb. For distance ratio q (i.e., q = distance to curb / distance to truck), let r(q) be the minimum speed ratio for which survival is possible. It turns out r(q) < 1, so r=1 is in fact always survivable. Under this speed ratio, survival occurs precisely at theta = arcsin(r(q)) (consistent with your model), which results in you reaching the curb at a vertical displacement equal to q * (initial distance to curb). So knowing only the distances rather than the speeds, it is possible to find the angle which is safe for the largest possible range of speeds. And if the truck speed happens to be below the highest survivable value, then the time-minimizing safe angle is somewhere in the range [0, theta). Anyway, none of this contradicts the analysis in your video, but there are definitely some unspoken assumptions in your translation of the scenario into an objective calculus question.
I love it! We further this analysis in two videos when we talk about the football pursuit problem. Those videos are here (links below), and we do a variation there that might be closer to your analysis. Or maybe yours is between our truck situation analysis and our football situation analysis. th-cam.com/video/5hk5bIEVVe8/w-d-xo.htmlsi=AQJJFqy5RAj7dPQC th-cam.com/video/4D-F2TwC9QU/w-d-xo.htmlsi=VSBGaE9_YmMFIa1u
@@MathTheWorld Your second video matches my analysis: the right angle strategy results in vertical displacement q*(initial distance to curb), by similar triangles. The isosceles triangle strategy in the first video resolves the "minimize time to curb among safe paths" problem, in the case of r=1, q >= 1. Of course, for r=1, q
I think the parameterization in these approaches still obscures the fundamental nature of the optimization problem. Let k be the speed of the car. Then let x be your horizontal velocity and k - y be your vertical velocity. Using D = RT, you know your hitting time will be inversely proportional to y, the relative speed of the car to you. In this same time period, your horizontal travel will be proportional to x. The ratio between horizontal travel with respect to the vertical starting distance is x/y. This ratio is what you want to maximize. Now imposing the speed condition gives x^2 + (k - y)^2 = r^2 for some speed threshold r. The problem reduces to finding the point of lowest angle on a circle of radius r centered at (0, k). This is given by the point of tangency, and the angle made with the y-axis will have a sine of r/k, since tangents are normal to radii.
Of course, this completely fails to account for any reaction by the driver. In reality, the average truck speed would decrease with larger angles, because the driver would have more time to slow down
Yep! We raise this in the video when we discuss whether Sara should err towards a smaller angle or a larger angle. But maybe we'll make another video where we include reaction time and deceleration in the model.
That and leaping is non-linear. I can dive five feet out of the way of a truck, but maybe not ten. So my first two steps are slower than my dive...but if I fail to dive far enough then my movement slows to almost zero.
The defensive back may not be able to use the formula but the ball carrier might, if we asume that the defender is the truck then the ball carrier needs to avoid the colision (obviating the fact that there are other people chasing you from behind)
Based on his calculus answer, the best amount is 10*sin(x), where x is arcsin(r) = arcsin(Sp/St). So this simplifies to 10*Sp/St = 10*(11.7/36.7) = 3.19, to three significant digits.
Bro... Nice video, but please use intelligent units! Or at least just show the metric units as well if you don't wanna use them... You talking about feet and using banana for scale systems just hurts to watch for all people that live in 99% of countries...
You make a good point. This was a compilation of three of our early videos, and we didn't know we would have such a big international following! Most further videos we do both, or just in metric. I do apologize.
I love this problem. I confess I ran out of time to finish watching the video (but I will watch it when I have more time, and perhaps edit or add to this comment). 1. Do you consider that the truck driver's choices will be affected by the pedestrians choices of path? In the given scenario it's reasonable to assume that they will probably maximize braking no matter what, but they may steer left or right based on where the pedestrian decides to go. If the pedestrian tries to turn around and go back, the driver should steer right. If the pedestrian tries to cross before the truck gets there, the driver should steer left. What are math problems like that called? 2. The pedestrian should not simply try to avoid a collision since that may not be possible. They should try to minimize the relative velocity at impact. In the scenario with the truck that makes traveling away from it instead of dodging it completely a reasonable option. 3. Perhaps the pedestrian should use a strategy which considers a probability of where the truck could travel. Again assuming maximum braking, the distance the truck travels can be predicted fairly accurately. There are also limits on how far left and right the driver could steer, as well as which direction they are likely to choose. The resulting probability should be used to guide which direction the pedestrian travels. I would choose a policy combining 2 and 3.
0:30 in this precise situation, going backwards (i.e. to the right) is likely optimal. Chances are that if there is a vehicle in the next lane, it is farther away from you. If it's not, then you may as well stand in the middle and hope both vehicles swerve a little bit. If the other vehicle is very small, jumping over it is more safe, even though you will probably still get injured. A moving vehicle can't hurt you very much if it's mostly under you.
What's funny is when I played football and we needed to Intercept the ball carrier, we were told to run at a 90 degree angle to Intercept them and never run at an angle or you'd over shoot. Worked every time.
Brilliantly done. But you should take the deceleration of the truck (and acceleration of Sarah) into consideration: at some point: the vehicle is gonna notice the pedestrian and do an emergency braking between 0.5 and 1 G
We sure hope so! But we focused on a situation that is more towards the worst-case scenario. It would be interesting to take it one step further and do what you suggest.
How do we know the fastest path is a straight line and not something curved like a brachistochrone? Would the optimal angle not change as we approach the curb or as t, time, increases? Also what about a problem where the truck slows down at a linear rate, or more likely, a non-linear rate?
Suppose some curved path lets you safely reach the curve. Let p be the point at which that curve intersects the curb, i.e. the point at which you become safe. A straight line to that point would also keep you safe, and would reach that point in less time.
As shown by the similar triangles diagram, if everything else stays the same, the optimal angle would not change. In the practical scenario, this means if the truck driver does not see you and continues at a constant speed, and similarly you are able to run at a constant speed from your starting point, then the optimal angle will not change. Given that the optimal angle changes with speed, the pedestrians optimal angle will change when the driver applies brakes. As the pedestrian could theoretically forecast their own running speed, I would suggest that the optimum angle should relate to average speed, given that a curving path increases the length of the path to the curb without as much of a corresponding increase to the distance from the truck, so we just need to think about the truck’s changing speed. To start off with, the pedestrian should start running at the optimal angle for the truck’s initial speed, presuming that the driver is not going to be able to put their brakes on (or not in time to make a significant difference). If the truck only just hits the breaks right at the last moment, the difference in its speed will be minimal, and this will likely be cancelled out by the reduction in speed possible when running on a curve compare to straight. If a driver was able to hit the breaks earlier, and still had a significant distance in which to slow down, the optimum angle to maximise the likelihood of getting to the curb (ie the ‘safety time’) would be an average of the truck’s speed across it’s deceleration. In the case where a driver sees a pedestrian, they are going to hit the breaks as hard as they can, and thus should have a linear deceleration, so a single optimal angle from the moment of breaking could theoretically be calculated. However, as a real pedestrian would be unlikely to be able to accurately forecast the deceleration power of a previously unknown truck, the best *course of action* might be to guess at an optimal angle in relation to the truck’s initial speed, and then essentially reassess as the truck slows down (if it in fact does) to check if a different angle would now be optimal. Changing to said new angle would involve something of a curve, as that’s how momentum impacts a turn, but the reason for the curve would be the transition from one angle to a different, more optimal angle. Again, running on a curve, (and recalculating an angle, even if it is not done in a particularly mathematical way) takes extra time, so it would not be most efficient to recalculate an optimum angle every split second and therefore take a curving path; and if this were possible, it would likely be possible to forecast at least some of the split second differences in speed and thus calculate an initial angle to reflect the end of the curve. I think. Don’t take my word for it, I’m only an English major.
The shortest distance does not correspond to the shortest time, except in special cases. Let's say you do tbe calculation lightning fast and head toward the location prescribed by the model. Since you're not capable of instantaneous acceleration, you must change your preexisting course to match your new one, describing a curve in the process. Special cases are when you're already moving in the right direction and when you start from a standstill.
Given Sarah's position and the truck position she should run back or never put herself in that initial position. Being in the habit of outrunnig trucks to cross the street is going to reduce her chances of survival to old age.
This is a nice expamle how you can use correct formulas and get wrong results due to wrong assumptions. Humans can accelerate crarzyly fast. If you are 10ft away from the curb, just turn around! Except for this solutions, the benefit of the other soluitons depend clearly from the response of the truck driver. in the original video, this only worked out, because the driver hit hit brakes.
There are two of us that work to make these videos. We are both Math teachers or former Math teachers. The person whose voice you hear on the video is a math education professor, and enjoys teaching college math classes like calculus and linear algebra. The person that does the visuals and editing, has a master's degree in math education and also years of experience in graphic design.
After using a bunch of linear algebra, a plethora of calculus, complex analysis, a pinch of quantum mechanics, trigonometry, involving the concept of butterfly effect and applying fluid dynamics to account for air resistance, drag and also brownian movement to evaluate precisely what angle is optimal and understanding the fundamental of minmax and I concluded any angle which does not heads you toward the other side will work, just stay and wait for the truck to go as you clearly mentioned distance to the other side is not minimised for optimal angle but distance from the truck to us to be maximised
9:28 the driver sees the child running and chooses a value from r in (15, inf) where r is the radius of the circle formed by the swerving cars part, and randomly chosen either right or left
If you are crossing the road, be sure you're able to see the next road's approaching car. In a one way road, after you drop from the bus go behind the bus to see the next road. In a two way road (where the next road goes opposite), go in front of the bus you just dropped (or wait for it to go) so you see the approaching car. If it's a 2 way road but the next road still goes along your path, go behind the bus like in the 1 way scenario. I almost had my life taken from me twice with motorcycles after I dropped off from my commute bus.
1:30 I'm pretty sure the clip of the bus and truck from the beginning of the video is from Norway (presumably near Oslo because it looks like the green Ruter buses we have here) so I'm pretty sure it's 3 meters rather than 10 feet 🤪
some of us check that the road is safe to cross before crossing. alternately, as the diagram was originally drawn, just stopping in place would allow the truck to pass w/o hitting. no calculus needed.
7:16 yhing isthe truck front is parallel to the crossing. Therefore your "x more feet" additional distance for the driver's to stop are actually lesser than that
This was a good video, I tried doing it before watching the video and over-complicated it by not realizing the relevant equation to maximize was the "safety time". I accidentally created an inequality that let me figure out the range of angles that let us get to the other side of the road, but had trouble generalizing it for the sports cases or using it in the case where it's impossible to cross safely.
Wow! I'm so impressed! Trying to solve problems with out an already known solution strategy is the number one key in learning to be a good problem solver, and learning mathematics as well! Researchers in math education call it productive struggle.
You can avoid 12 minutes and look from the perspective of the truck driver and the answer is trivial for uniform movement. This works even for a real case like in the video at the start, where both the truck breaks and the kid accelerate. It looks that the kid that survived and the driver did better than this math, none of them went on a straight line.
Thanks. Now I know to always bring a calculator, and a speed gun when crossing the road. edit: And a protractor and pen to measure on the floor. edit: And a ruler to put perpendicular to the side of the road so I can use the protractor.
Or you could just look both ways before crossing the road to make sure you don’t end up in the path of a truck in the first place Either that or as you start to enter the path of the truck, you could just run back the way you came to avoid the truck because that would be much less distance to cover
Using pain in the ass units is a good strategy to encourage weaning off the non-calculus solutions, and get to the platonic pure beautiful dimensionless answer ;)
We need more problems of the real world optimised using calculus. About the problem, at first, even knowing calculus very well, I thought the distances and the speeds would matter quite a lot, but in fact the distances don't matter to calculate which angle is the most optimal, it's just about whether or not you're lucky enough to make it alive. And the speeds only matter in direct proportion to each other, which is very neat. But still, if I am in a desperate situation like this, and I can not at all estimate the ratio of our speeds, where should I run? What I'm getting is that, if you're SIGNIFICANTLY slower you should run straight to the edge, since that's both optimal and the driver can't stop in time, but if you are decently slower but not too much (like say the car is twice or three times as fast, like usually in a city), a slight turn is good, and overshooting slightly is not particularly harmful since you give the driver time. These are, as I said at the very beginning, the things that are genuinely helpful but extremely hard to get across.
A driver's initial reaction is to turn away from you, so that would be towards the curb, in which case your best bet would be to stop and/or move backwards.
That actually adds another reason to go away from them, if you do end up going to the curb. It gives them more space to turn and get out of their way. I have wondered at what point is it safer for you too stop and turn around, than to keep going forward because that's where your momentum is taking you.
All right here's my favourite one... Say you're some distance away from a river, willing to get some water home. You see some fire at the same side of the river. At what point on the river should you run to minimise your time to reach the fire with your bucket of water? (PS its a numberphile video problem kinda twisted with words)
A fun problem. I'm reminded of those silly cartoon where a tree or building is falling towards the character. They always stupidly start running directly away from the falling tree, hoping to escape. But they are unable to run the length of the tree/ building before it falls on them. I find myself always shouting, "Run across, don't run directly away!!" lol But I suppose some angle would be optimal.
I thinl proportional guidance is probably more relevant for the football case due to trying to intercept something than dodging something at fixed directions
So I'm curious now... is the best path actually a straight line or a curve? IE, perhaps it's better to start running toward the curb initially and then rotate to run away from the truck as you get closer to the curb, both minimizing your time to get to a safe location and maximizing your time to impact. This might take acceleration into account.
It is a straight line, at least with our assumptions. We have another video that focuses on a pursuit problem in American football. In that video we developed a rule of thumb that allows you to find a possible path to survive, if there is a possible path that exists. The strategy is to go 90° from you and the player downfield. Or in this case, go perpendicular to the segment that connects the far corner of the truck and you. Here is the link: th-cam.com/video/TnS02YZoygc/w-d-xo.htmlsi=ApLGjX-dBBXkHBgP
This is the kind of problem i need on my math test
Yea like where u have to to shoot an arrow to hit a moving target
Is this not the kind of problem that you have on your math tests?
@@hisanuswat4359i've heard people get shit like this instead:
"There are 2 trains about to collapse into each other at X speed, how fast does it takes for a fly in X speed to touch both of them before they clash ?"
Not entirely correct to the actual question, but something similar.
@@dev4159 I guess it's similar. I mean, I myself have never found exactly this problem on tests, but similar ones, but we usually got such problems in vectors and 2 dimensional motion stuff
This is actually far easy ...@@dev4159
Next time you make a video on how to not get hit by a car, could you make it a bit shorter? The car is getting closer and I still have several minutes left in the video
Ahhh! You want a TH-cam short, that would make a lot of sense!
underrated comment
I can tell you now so you don’t have to wait for the video to be over
Don't worry you have to wait until the truck is exactly 40ft away to start running or the math doesn't work
Better question: If, hypothetically, there was a truck trying to hit a person, what is the path they must take to most optimally try to hit them given that the person is taking the most optimal path to run away?
I would say the other question is a lot more likely but this is a good one too
How about: If, hypothetically, there was a truck trying not to hit a person, what is the path they must take to not run into the path the truck is swerving to not hit them?
Turn 90 degrees
Go *straight* to jail. Do not pass go!
You are asking for classified and export controlled information.
Please turn yourself in to the nearest USAF base immediately.
No it doesn't matter that you only asked about a 2D representation of terminal guidance that has been public knowledge for more than half a century.
Do a drift 🧠🔥
When I was in high school, we started learning about polar functions, including limaçons. Separately, I noticed that when an overhead light shined on a mug of milk, the light made a pattern on the milk that looked suspiciously like a limaçon. I was able to prove it, and I challenge Math The World to do the same!
challenge accepted! Though it may be a minute before we get to it, just fair warning 😉
I think there’s already a video on that by Minute Earth
The "Caustic (optics)" wiki article also has some cool info on a related topic.
What's liçma?
@@Jabberwockybirdlicma balls
PLEASE use the metric system
Yes lol, it is very annoying when someone uses "feet" "miles" "Fahrenheit" or something... I have to convert them to metric system just to understand them
"Let's convert mph to fps."
How about NO?
I don't see why you feel the need to convert the units to metric. The units don't matter. It works exactly the same no matter what units you use. He might as well have used furlongs per fortnight or potrzebies per clarke. Lightyears per plank time, maybe.
Also, when the units do matter we feel the same way about someone using meters, °C, etc. We have to convert to feet, °F, etc. to understand. Why would I make my problem not understandable to myself to make it understandable to some random person on the internet? That's what truly doesn't make sense.
@@pjl22222 absolutely nobody asked
Just go backwards
You didn’t have to cut me off
True, but turning around and changing direction takes time
Make out like it never happened ‼️🗣️🗣️🗣️🧠🧠🧠💯💯💯
@@Somebody71828 And I don't even need your love
@@WhirlwindQuestEver seen moonwalking? Spoiler alert: you do not need to turn around to walk back. Honestly, jumping backwards is probably the fastest dodge.
I am the Sara in this problem! Happy I got to make a guest appearance!
actually?? no way 😮
I will vouch for her. That is my one and only daughter.
@@MathTheWorld 😲
It makes sense that it would be based on the ratio of speeds, since changing both speeds by the same factor is equivalent to running an identical scenario in either slow motion or fast forward.
I like the logic! Thanks for sharing
That’s a fair point, but I would add that it’s still somewhat enlightening that the solution turns out to be ONLY dependent on that ratio. My gut assumption was that the ratio of speeds would be the biggest contributor, but the distance from the curb and the distance from the truck would also matter. (They do, but only in the binary sense of “will you make it or not?”)
3:13 **laughs in metric**
fr
Exactly what I was thinking HAHAHA
Please help me, I'm a little confused. Which power of 10 do I multiply by to convert kph to m/s?
@@pjl22222 1 m/s = 3.6 km/h and the ratio 3.6 comes from 3600/1000 (number of seconds in 1h)/(number of meters in 1km) so it's not a power of 10 but 3.6
@@pjl22222 it’s easier to look at it this way: imagine you have to convert x km/h to m/s. x km/h = x km/h*1*1= x km/h * 1000 m/km * 1h/3600s Now the km and h cancel out leaving you with x*1000/3600 m/s = x*10/36 m/s = x*5/18 m/s
Cool video, but the fastest way to go is backwards. Made even faster because trying to turn takes time, while when going backwards you don’t need to turn at all.
Actually, it is a good question about when is it quicker to go backwards and when is it better to go forward. You already have momentum going forward, so presumably, in the same time period, you could go further going forward than backwards, which would require turning or stopping/walking backwards.
There is also the issue with the traffic in the lane behind you that you also have to avoid.
@@MathTheWorld Yeah I forgot that there would be traffic in other lanes.😅
@@ExzaktVid Many years back, I had an accident cycling through an intersection with a car crossing. Minimal background info: I have no 3d sight, so am bad at guessing distances and speeds. Experience helps, so long as situations are consistent with speeds - turns out (according to other people), the car was speeding.
The problem I had was, essentially, indecisiveness - I changed several times my opinion if I should try to beat the car, or if it would be better to stop. turns out the result was a collision where I was pretty much unhurt and the bike pretty much undamaged - no real idea about the car, because by the time it managed to stop, it was a good bit beyond the intersection and I couldn't see the sides. The driver obviously thought they were not free of guilt, from them only ever asking if I was ok.
Obviously, what would have been right depends on speeds and positions ... which I wasn't best at guessing, but the math might still be interesting.
Go backwards and stay in the middle of the road (one with the lines)
Again, the backward strategy is just assuming you don't already have strong forward momentum, otherwise you would just slide across the concret (and likely fall too)
1:19 bro got 8 kids?! Was he trynna do a math problem with his wife 😭?
he trying to calculate most optimal number of kids for his family lol
This deserves more likes
Multiplication
in theory the Truck should be trying to slow down (that's also what saved that kid in the opening video)
Could try to explore what would be happening with a truck speed reducing, but not enough to totally be at rest before meeting the perpendicular.
Nice video otherwise. That's a very good example of Calculus power !
Yes we definitely took a worse case scenario here of if the truck never saw you and continues at its speed! But if it slowed down then that's even better for Sara!
@@MathTheWorldit's even better for Sara, but that may mean there's an even larger benefit for moving away from the truck, because the truck would be going slower the more you increase the distance away from the truck. It may make the optimal angle steeper, but I do think the math gets much harder at that point
Yeah, when this situation happens, I'd advise Sara to inquire about the breaking capabilities of the specific type of truck from the manufacturer first, and to also weight in the degradation of the specific truck's breaks to come up with the optimal solution. X)
But to be real, the angles are likely a lot different when considering breaking, especially if we consider that it doesn't matter too much which part of the truck hits us (as long as the front is flat), but it matter a lot, how fast it hits us.
So if you just cannot avoid being hit, then it may be worth to run straight away, so the truck causes the least damage to you (not only due to breaking, but also because your speed is also subtracted from the truck's "damaging potential" when the hitting happens)
@@MathTheWorldto do this you could use integration. If the velocity of the truck is defined by v0-x (v0=initial velocity), its integral, and therefore its position, would be defined by s0+v0*x-(x^2)/2. I believe this would greatly change the angle. Im not going to follow through on the math right now, but I might some other day!
@@MathTheWorld Which is good because if the truck sees you the analysis gets really complicated with how the driver responds to seeing you and how that affects the optimal route.
Favorite math problem.🤔
The is tuff. I love linear algebra. But the thing i use the most to make sense of the world are simple energy analisys, simple heat flow and unit analysis.
Combined with +-*/ and % calculations you can get a solid understanding or approximation of almost any day to day problem.
right now my favorite math problems are circuit analysis, and my understanding is that they're just one part of heat analysis and thermodynamics stuff
@@janthran I have only done very simple steady state circuit analysis many years ago, so I don't know how alike they are.
But they at least can have the steady state assumption in common.🤓
The answer is backwards, out of the way of the truck and waiting until it is past.
Boom! No math needed lol. 😄.
This is a really interesting video though.
I apologize if I came across as argumentative, I was trying to be humorous, but I'm autistic and sometimes things don't come across as I intend them.
Yeah, we worried viewers might think that with the position we drew the person. We should have drawn another vehicle coming up the other lane, so then that option is even more dangerous than going forward.
It certainly it is a valid option. I wonder at what point is it better to turn around/stop and go backwards vs going forward, since you already have momentum going forward. I don't know if a mathematical model would give you a very good estimate here, but it is an interesting question.
haha. Yeah. I was going to say, step backwards LoL😮😅😊
@@MathTheWorld So, Sarah should just stay between the lines
Turning around is usually what gets squirrels flattened
@@pjl22222 yes, turning around is what causes squirrels to get flattened. But humans have the ability to move backwards without having stop and turn around. It's one of the benefits of being bipedal.
This is THE BEST VIDEO explaining what are the the differences between different math levels and WHY we invented calculus. THANK YOU SO MUCH. I will share this video a lot
Thank you so much! Please be sure to share it with any Math teachers that you know, if you're willing.
The bigger the truck is, the harder it is to estimate its speed, the bigger the uncertainty of that factor.
If you run straight away from the truck, then you also minimize speed difference. Let's say your running speed is 10 km/h and the truck comes incredibly fast and breaks incredibly fast, so that you have 0 chance to avoid being hit and your angle won't significantly change the time when you get hit or the speed of the truck at the moment of collision, but because of incredible strong breaks the trucks speed at that point in time is just 15 km/h. So if you run straight away from it, then you get hit with a relative speed of just 5 km/h, which you are very likely to survive. If the trucker releases the breaks as soon as he reaches 5 km/h, then you'll get run over slowly and still die, but let's not assume that.
If you run perpendicular, then the relative speed is sqr(10²+15²)≈18km/h which is almost impossible to survive. In fact if the truck driver didn't buckle his seat belt and you're a giant block of concrete, then there's a pretty good chance that the truck driver dies. Ironically in this case (minus the concrete block thing) your chances of survival would be marginally bigger if you just stood still. Don't do that though.
My solution: dont cross the street without taking a good look and take a break for a few seconds and then cross. Also dont cross between other cars when you cant see the other lane. Again take a break and look before crossing
Yes, this is the best solution. Your point about crossing when you can't see both lanes (or all 4 lanes on a larger street) is so important. We had a couple of kids killed here in Utah a few years ago because of a well-meaning driver stopping on a 4 lane road to let the kids cross in front of his/her car. Well, his/her big SUV blocked the view of other drivers in the lane next to him/her, and couldn't see the kids, and the kids couldn't see the car in the other lane either. As soon as they cleared the SUV and got into the second lane, they were immediately hit by a car. It was so sad.
Why are so many people complaining about the units 💀
I’m a metric guy, but the units used have absolutely 0 bearing on the underlying math or visualizations. He could’ve even not used any system and just said “units per time interval” and it’d all still make perfect sense
what in the american units is this video...
unfortunatelly
It’s all good 🎺
WHAT THE FUCK IS A KILOMETER 🇺🇸🇺🇸🇺🇸🇺🇸🇺🇸🇺🇸🦅🦅🦅🦅🔫🔫🔫🔫🔫🍔🍔🍔🍔🍔
Truck speed in ft/s is why I'm about to skip the rest of the video...
I think about this everytime I cross the street. Thanks for this
I just want to let you know that I love this channel so much and hope you never stop
Thank you so much! We plan on going for a long time!
Great video. I think one thing that might help the viewer look for insights would be to highlight that even before you pointed it out at 9:05, we already knew that it doesn't matter what the actual speeds are, because we know that the actual distances don't matter.
If it doesn't matter that the truck is twice as far away, or that the curb is closer or further, then it stands to reason that the actual speed doesn't matter.
By the time I completed all the calculations.. I was roadkill...
Before calculating the optimal angle of escaping the truck, lets first calculate how 3 dimensional we have to be to complete the calculation.
@@somerandomdragon558 For dimensional actually.. 3d space 1d time..
I remember wondering about this on my way to school, but never bothered to figure it out. This has given me the motivation to do that
bro rly just fermi estimated and made it look simple
It's not clear to me that maximizing distance between you and the truck at the curb is the right thing to optimize, as opposed to say, minimizing the time to the curb among all paths which keep you safe. And as you noticed, this model behaves weirdly when r is at least 1. Sure, it's trivial to survive when r > 1, but if r = 1, this model prescribes running parallel to the curb forever, and fails to ascertain whether there is any safe way to reach the curb.
For distance ratio q (i.e., q = distance to curb / distance to truck), let r(q) be the minimum speed ratio for which survival is possible. It turns out r(q) < 1, so r=1 is in fact always survivable. Under this speed ratio, survival occurs precisely at theta = arcsin(r(q)) (consistent with your model), which results in you reaching the curb at a vertical displacement equal to q * (initial distance to curb). So knowing only the distances rather than the speeds, it is possible to find the angle which is safe for the largest possible range of speeds. And if the truck speed happens to be below the highest survivable value, then the time-minimizing safe angle is somewhere in the range [0, theta).
Anyway, none of this contradicts the analysis in your video, but there are definitely some unspoken assumptions in your translation of the scenario into an objective calculus question.
I love it! We further this analysis in two videos when we talk about the football pursuit problem. Those videos are here (links below), and we do a variation there that might be closer to your analysis. Or maybe yours is between our truck situation analysis and our football situation analysis.
th-cam.com/video/5hk5bIEVVe8/w-d-xo.htmlsi=AQJJFqy5RAj7dPQC
th-cam.com/video/4D-F2TwC9QU/w-d-xo.htmlsi=VSBGaE9_YmMFIa1u
@@MathTheWorld Your second video matches my analysis: the right angle strategy results in vertical displacement q*(initial distance to curb), by similar triangles. The isosceles triangle strategy in the first video resolves the "minimize time to curb among safe paths" problem, in the case of r=1, q >= 1. Of course, for r=1, q
I think the parameterization in these approaches still obscures the fundamental nature of the optimization problem. Let k be the speed of the car. Then let x be your horizontal velocity and k - y be your vertical velocity. Using D = RT, you know your hitting time will be inversely proportional to y, the relative speed of the car to you. In this same time period, your horizontal travel will be proportional to x. The ratio between horizontal travel with respect to the vertical starting distance is x/y. This ratio is what you want to maximize. Now imposing the speed condition gives x^2 + (k - y)^2 = r^2 for some speed threshold r. The problem reduces to finding the point of lowest angle on a circle of radius r centered at (0, k). This is given by the point of tangency, and the angle made with the y-axis will have a sine of r/k, since tangents are normal to radii.
Thanks for the video!
I actually liked that you used the units you did. It makes it more applicable to me.
Of course, this completely fails to account for any reaction by the driver. In reality, the average truck speed would decrease with larger angles, because the driver would have more time to slow down
Yep! We raise this in the video when we discuss whether Sara should err towards a smaller angle or a larger angle. But maybe we'll make another video where we include reaction time and deceleration in the model.
That and leaping is non-linear. I can dive five feet out of the way of a truck, but maybe not ten. So my first two steps are slower than my dive...but if I fail to dive far enough then my movement slows to almost zero.
The defensive back may not be able to use the formula but the ball carrier might, if we asume that the defender is the truck then the ball carrier needs to avoid the colision (obviating the fact that there are other people chasing you from behind)
This is the video egyptians crossing highways must have watched to get so good
I'm at 3:45, and I would say it's barely over 3 feet, so I'd guess more like 3.1 feet is the best amount.
Based on his calculus answer, the best amount is 10*sin(x), where x is arcsin(r) = arcsin(Sp/St). So this simplifies to 10*Sp/St = 10*(11.7/36.7) = 3.19, to three significant digits.
you don't need music. your narration and visuals are good enough.
This is a prime example of why being good at math ≠ having practical common sense in real life.
Great video. I'd love to see further analysis that adds in reaction time and decelleration of the truck as parameters.
The answer is up, drink redbull to grow a pair of wings and fly and now you'll avoid getting hit by the majority of cars!
Bro... Nice video, but please use intelligent units! Or at least just show the metric units as well if you don't wanna use them... You talking about feet and using banana for scale systems just hurts to watch for all people that live in 99% of countries...
You make a good point. This was a compilation of three of our early videos, and we didn't know we would have such a big international following! Most further videos we do both, or just in metric. I do apologize.
This is a very well produced and explained education video; you show a multiple level approach to tackling a complex problem.
Thank you very much!
I love this problem.
I confess I ran out of time to finish watching the video (but I will watch it when I have more time, and perhaps edit or add to this comment).
1. Do you consider that the truck driver's choices will be affected by the pedestrians choices of path? In the given scenario it's reasonable to assume that they will probably maximize braking no matter what, but they may steer left or right based on where the pedestrian decides to go. If the pedestrian tries to turn around and go back, the driver should steer right. If the pedestrian tries to cross before the truck gets there, the driver should steer left. What are math problems like that called?
2. The pedestrian should not simply try to avoid a collision since that may not be possible. They should try to minimize the relative velocity at impact. In the scenario with the truck that makes traveling away from it instead of dodging it completely a reasonable option.
3. Perhaps the pedestrian should use a strategy which considers a probability of where the truck could travel. Again assuming maximum braking, the distance the truck travels can be predicted fairly accurately. There are also limits on how far left and right the driver could steer, as well as which direction they are likely to choose. The resulting probability should be used to guide which direction the pedestrian travels.
I would choose a policy combining 2 and 3.
Now I'm afraid to go to my math test for fear I'll be run over by a car on the way. I should have done more studying and less partying last week.
0:30 in this precise situation, going backwards (i.e. to the right) is likely optimal. Chances are that if there is a vehicle in the next lane, it is farther away from you. If it's not, then you may as well stand in the middle and hope both vehicles swerve a little bit. If the other vehicle is very small, jumping over it is more safe, even though you will probably still get injured. A moving vehicle can't hurt you very much if it's mostly under you.
1:19 8 kids are you serious doing some long division and multiplication with the wife this guy really does love math
In reality the optimal angle is 180º.
Do not run 180° away from a semi truck going at a constant speed, worst mistake of my life
What's funny is when I played football and we needed to Intercept the ball carrier, we were told to run at a 90 degree angle to Intercept them and never run at an angle or you'd over shoot. Worked every time.
Instructions unclear; the truck ran over my calculator
Brilliantly done.
But you should take the deceleration of the truck (and acceleration of Sarah) into consideration: at some point: the vehicle is gonna notice the pedestrian and do an emergency braking between 0.5 and 1 G
We sure hope so! But we focused on a situation that is more towards the worst-case scenario. It would be interesting to take it one step further and do what you suggest.
@@MathTheWorld A step further...in the right direction!
How do we know the fastest path is a straight line and not something curved like a brachistochrone? Would the optimal angle not change as we approach the curb or as t, time, increases? Also what about a problem where the truck slows down at a linear rate, or more likely, a non-linear rate?
Suppose some curved path lets you safely reach the curve. Let p be the point at which that curve intersects the curb, i.e. the point at which you become safe. A straight line to that point would also keep you safe, and would reach that point in less time.
As shown by the similar triangles diagram, if everything else stays the same, the optimal angle would not change. In the practical scenario, this means if the truck driver does not see you and continues at a constant speed, and similarly you are able to run at a constant speed from your starting point, then the optimal angle will not change. Given that the optimal angle changes with speed, the pedestrians optimal angle will change when the driver applies brakes. As the pedestrian could theoretically forecast their own running speed, I would suggest that the optimum angle should relate to average speed, given that a curving path increases the length of the path to the curb without as much of a corresponding increase to the distance from the truck, so we just need to think about the truck’s changing speed. To start off with, the pedestrian should start running at the optimal angle for the truck’s initial speed, presuming that the driver is not going to be able to put their brakes on (or not in time to make a significant difference). If the truck only just hits the breaks right at the last moment, the difference in its speed will be minimal, and this will likely be cancelled out by the reduction in speed possible when running on a curve compare to straight. If a driver was able to hit the breaks earlier, and still had a significant distance in which to slow down, the optimum angle to maximise the likelihood of getting to the curb (ie the ‘safety time’) would be an average of the truck’s speed across it’s deceleration. In the case where a driver sees a pedestrian, they are going to hit the breaks as hard as they can, and thus should have a linear deceleration, so a single optimal angle from the moment of breaking could theoretically be calculated. However, as a real pedestrian would be unlikely to be able to accurately forecast the deceleration power of a previously unknown truck, the best *course of action* might be to guess at an optimal angle in relation to the truck’s initial speed, and then essentially reassess as the truck slows down (if it in fact does) to check if a different angle would now be optimal. Changing to said new angle would involve something of a curve, as that’s how momentum impacts a turn, but the reason for the curve would be the transition from one angle to a different, more optimal angle. Again, running on a curve, (and recalculating an angle, even if it is not done in a particularly mathematical way) takes extra time, so it would not be most efficient to recalculate an optimum angle every split second and therefore take a curving path; and if this were possible, it would likely be possible to forecast at least some of the split second differences in speed and thus calculate an initial angle to reflect the end of the curve.
I think.
Don’t take my word for it, I’m only an English major.
The shortest distance does not correspond to the shortest time, except in special cases. Let's say you do tbe calculation lightning fast and head toward the location prescribed by the model. Since you're not capable of instantaneous acceleration, you must change your preexisting course to match your new one, describing a curve in the process. Special cases are when you're already moving in the right direction and when you start from a standstill.
I'm procrastinating on studying for my calculus exam by watching this video. The irony is palpable.
At 3:43 why set a constant value for "additional time for truck"? I expected that column to be A/36.7
yeah same. is this incremental? but the pythagorean is not linear, so it doesnt make sense to be incremental...
Loved it.
This channel is a goldmine
So I'm not crazy
No, I have also thought about this a lot
actually, you still may be😅😊
Crazy? I was crazy once. They put me on a road. A hypothetical road. A hypothetical road with maths. Maths makes me crazy.
Given Sarah's position and the truck position she should run back or never put herself in that initial position. Being in the habit of outrunnig trucks to cross the street is going to reduce her chances of survival to old age.
Friend: what is he doing, his been standing there for 15 minutes!
*(JUST CROSS THE ROAD!)*
What program do you use for drawing? It looks so nice. Also great video!
We use photoshop! We draw on the canvas and screen record as we draw
This is a nice expamle how you can use correct formulas and get wrong results due to wrong assumptions. Humans can accelerate crarzyly fast. If you are 10ft away from the curb, just turn around! Except for this solutions, the benefit of the other soluitons depend clearly from the response of the truck driver. in the original video, this only worked out, because the driver hit hit brakes.
Nice work. You should be a math teacher if you're not!
There are two of us that work to make these videos. We are both Math teachers or former Math teachers. The person whose voice you hear on the video is a math education professor, and enjoys teaching college math classes like calculus and linear algebra. The person that does the visuals and editing, has a master's degree in math education and also years of experience in graphic design.
Optimization problems are the best.
After using a bunch of linear algebra, a plethora of calculus, complex analysis, a pinch of quantum mechanics, trigonometry, involving the concept of butterfly effect and applying fluid dynamics to account for air resistance, drag and also brownian movement to evaluate precisely what angle is optimal and understanding the fundamental of minmax and I concluded any angle which does not heads you toward the other side will work, just stay and wait for the truck to go as you clearly mentioned distance to the other side is not minimised for optimal angle but distance from the truck to us to be maximised
9:28 the driver sees the child running and chooses a value from r in (15, inf) where r is the radius of the circle formed by the swerving cars part, and randomly chosen either right or left
I actually asked myself this question
Thanks for this video
as a proud US patriot, it would be easier to follow if you converted all units to hotdog per eagle. thanks! 🇺🇸 (sarcasm)
If you are crossing the road, be sure you're able to see the next road's approaching car.
In a one way road, after you drop from the bus go behind the bus to see the next road.
In a two way road (where the next road goes opposite), go in front of the bus you just dropped (or wait for it to go) so you see the approaching car.
If it's a 2 way road but the next road still goes along your path, go behind the bus like in the 1 way scenario.
I almost had my life taken from me twice with motorcycles after I dropped off from my commute bus.
1:30 I'm pretty sure the clip of the bus and truck from the beginning of the video is from Norway (presumably near Oslo because it looks like the green Ruter buses we have here) so I'm pretty sure it's 3 meters rather than 10 feet 🤪
Why doesn’t she turn around
some of us check that the road is safe to cross before crossing. alternately, as the diagram was originally drawn, just stopping in place would allow the truck to pass w/o hitting. no calculus needed.
Just when you think the video is done, he includes new variables to solve for
I think this problem has gone through my mind already because of that truck clip
7:16 yhing isthe truck front is parallel to the crossing. Therefore your "x more feet" additional distance for the driver's to stop are actually lesser than that
You know your mathematician father loves you when he puts you in a hypothetical risk of dying and hypothetically saves you with math
Yes, that is true fatherly love.
This was a good video, I tried doing it before watching the video and over-complicated it by not realizing the relevant equation to maximize was the "safety time".
I accidentally created an inequality that let me figure out the range of angles that let us get to the other side of the road, but had trouble generalizing it for the sports cases or using it in the case where it's impossible to cross safely.
Wow! I'm so impressed! Trying to solve problems with out an already known solution strategy is the number one key in learning to be a good problem solver, and learning mathematics as well! Researchers in math education call it productive struggle.
Fun fact, finding the optimal angle to run at was actually a question in the singapore physics league (sphl)
I want to say going backwards but who knows what cars are behind you or are coming on that side of the road
You can avoid 12 minutes and look from the perspective of the truck driver and the answer is trivial for uniform movement. This works even for a real case like in the video at the start, where both the truck breaks and the kid accelerate.
It looks that the kid that survived and the driver did better than this math, none of them went on a straight line.
This gonna be on my homework one day isn't it 😭
Super interesting analysis! Just commenting for the algorithm, ha
We appreciate it!
I’m Canadian and we use both Metric and Imperial and Metric is like a universe ahead
Since in the question asked about a TRUCK, not any vehicle, I think the correct answer is dive straight downwards and go prone between the wheels
Thanks. Now I know to always bring a calculator, and a speed gun when crossing the road.
edit: And a protractor and pen to measure on the floor.
edit: And a ruler to put perpendicular to the side of the road so I can use the protractor.
Or you could just look both ways before crossing the road to make sure you don’t end up in the path of a truck in the first place
Either that or as you start to enter the path of the truck, you could just run back the way you came to avoid the truck because that would be much less distance to cover
1:18 bro has 8 children
I would have really liked to see you bring partial differential equations into this.
Damn, you brought math into this...
This is sick
REALLY VERY UNDERRATED CHANNEL
Thank you!
No, only if he would use actually the metric system.
No idea, why would a mathematician use feet or miles...
Using pain in the ass units is a good strategy to encourage weaning off the non-calculus solutions, and get to the platonic pure beautiful dimensionless answer ;)
We need more problems of the real world optimised using calculus.
About the problem, at first, even knowing calculus very well, I thought the distances and the speeds would matter quite a lot, but in fact the distances don't matter to calculate which angle is the most optimal, it's just about whether or not you're lucky enough to make it alive. And the speeds only matter in direct proportion to each other, which is very neat.
But still, if I am in a desperate situation like this, and I can not at all estimate the ratio of our speeds, where should I run? What I'm getting is that, if you're SIGNIFICANTLY slower you should run straight to the edge, since that's both optimal and the driver can't stop in time, but if you are decently slower but not too much (like say the car is twice or three times as fast, like usually in a city), a slight turn is good, and overshooting slightly is not particularly harmful since you give the driver time.
These are, as I said at the very beginning, the things that are genuinely helpful but extremely hard to get across.
A driver's initial reaction is to turn away from you, so that would be towards the curb, in which case your best bet would be to stop and/or move backwards.
That actually adds another reason to go away from them, if you do end up going to the curb. It gives them more space to turn and get out of their way.
I have wondered at what point is it safer for you too stop and turn around, than to keep going forward because that's where your momentum is taking you.
neat !
I loved this video
Thanks!
looks like the questions of walking vs running in the rain :)
All right here's my favourite one... Say you're some distance away from a river, willing to get some water home. You see some fire at the same side of the river. At what point on the river should you run to minimise your time to reach the fire with your bucket of water? (PS its a numberphile video problem kinda twisted with words)
really fun solution... thats makes you say AHHHH!! wow! nice!
what video is it? hard to find with just this
Can you do something similar but use the example from the movie Prometheus when the main character was running away from a rolling ship?
The optimal angle in that sort of situation would be 180 degrees, that is, to stop and walk backwards.
1:39 "What point on the curb should Sarah run to?"
Actually in this situation, Sarah should just take one step backwards 🤣
Sara is clearly safe right where she's at, between the lanes, the correct response is not to move at all.
A fun problem. I'm reminded of those silly cartoon where a tree or building is falling towards the character. They always stupidly start running directly away from the falling tree, hoping to escape. But they are unable to run the length of the tree/ building before it falls on them. I find myself always shouting, "Run across, don't run directly away!!" lol But I suppose some angle would be optimal.
no way, I've literally thought about this exact question many times
I thinl proportional guidance is probably more relevant for the football case due to trying to intercept something than dodging something at fixed directions
0:35 I feel like jumping right would be better
So I'm curious now... is the best path actually a straight line or a curve? IE, perhaps it's better to start running toward the curb initially and then rotate to run away from the truck as you get closer to the curb, both minimizing your time to get to a safe location and maximizing your time to impact. This might take acceleration into account.
It is a straight line, at least with our assumptions. We have another video that focuses on a pursuit problem in American football. In that video we developed a rule of thumb that allows you to find a possible path to survive, if there is a possible path that exists. The strategy is to go 90° from you and the player downfield. Or in this case, go perpendicular to the segment that connects the far corner of the truck and you. Here is the link:
th-cam.com/video/TnS02YZoygc/w-d-xo.htmlsi=ApLGjX-dBBXkHBgP
I met this problem once at the county-level physics olympics
... I didn't advance to the next stage