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

Ahhh! You want a KZfaq 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

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 🧠🔥

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

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

​@@GoatStormChaserEver seen moonwalking? Spoiler alert: you do not need to turn around to walk back. Honestly, jumping backwards is probably the fastest dodge.

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

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?”)

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

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)

he trying to calculate most optimal number of kids for his family lol

This deserves more likes

Multiplication

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.

actually?? no way 😮

I will vouch for her. That is my one and only daughter.

@@MathTheWorld 😲

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.

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.🤓

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.

Thank you so much! Please be sure to share it with any Math teachers that you know, if you're willing.

Thank you so much! We plan on going for a long time!

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.

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. kzfaq.info/get/bejne/a86baMV6qry6lWw.htmlsi=AQJJFqy5RAj7dPQC kzfaq.info/get/bejne/aqpdeZWF3KmdgYk.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

pi!

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.

We use photoshop! We draw on the canvas and screen record as we draw

yeah same. is this incremental? but the pythagorean is not linear, so it doesnt make sense to be incremental...

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.

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..

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.

Do not run 180° away from a semi truck going at a constant speed, worst mistake of my life

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.

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.

We appreciate it!

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 ;)

Thanks!

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: kzfaq.info/get/bejne/itSDY5WKv9Xdl5c.htmlsi=ApLGjX-dBBXkHBgP

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!

Great question! We actually do have a video about that, but we do it for a water bottle instead of a glass/cup. We did this problem so people would know how much water to put in the water bottle to make it easier to do a bottle flip. kzfaq.info/get/bejne/rdWVe9Rmtc3eZHk.htmlsi=hBwcbhXO46VLHZqA

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.

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.

Nice intuition! The angle of refraction of light in different mediums and stuff using the same formula we develop here with the arcsine of the ratio of the speeds.

You're right. There are multiple things missed here, including (correct me if I'm wrong) the additional truck time for truck at 3:52, which should vary based on added feet but doesn't

Actually, why don't you help me and tell me how you got going that fast! Of course at that speed surviving a truck collision is the worst of your worries. Check out the "What If" chapter on throwing a baseball at 90% the speed of light.

That would be an interesting question! And if the truck was decelerating at a constant rate, how would that change it? We might have to make a follow-up video.

really fun solution... thats makes you say AHHHH!! wow! nice!

what video is it? hard to find with just this

That is the additional time for each additional foot down the curb that Sara goes. You need to sum them up to get the total additional time.

@@MathTheWorld ah i see! thank you for taking the time to reply to me!