@DrTrefor
Make sure you check out the many other cool videos using the #SoME2 hashtag, so many good ones! I had so much fun making this video for the Summer of Math Exposition 2, thanks to everybody who has been part of this experience:) Errata: 1) at 5:33 the +L in the assumption should be a -L
@absolutelyuniformlyconfuse4176
Really fun video! I feel like mathematical modeling is something that doesn’t get enough respect in the online math community. I personally did a “math modeling competition” (specifically the “M3 challenge”) recently and was shocked at how it’s in many ways a completely different skill set than solving a pure math problem, and in many ways I found it significantly harder.
@johnchessant3012
This is really cool! I like how this gives some variety for people like me who are more pure math inclined, to see from start to finish an example of a model studying something in the real world. Definitely wish I had done at least a little bit of that in school!
@Salien1999
I found this video really interesting! Towards the tail end of my degree in mechanical engineering, I actually took a grad course in modeling energy systems and the methodology was very similar to what you describe in the video. The only difference was (because we're engineers, and not necessarily mathematicians) once the math got to the point of having to solve complex systems of differential equations, we typically defaulted to using some sort of numerical method to approximate the solution and develop a program to get a solution (much like you did with MATLAB, except we probably would've done it much sooner. Or I would've. My math skills have waned over the years, admittedly). Only thing I can add for discussion, which I can't remember if you discussed in the video or not: once you reach the limits of your ability to exactly solve the differential equations that "exactly" model reality, there are two approaches: find an exact solution to an approximate model (simply the equations until you can solve them to get an exact numerical answer to equations which may or may not reflect reality), OR find an approximate answer to a more exact model (use numerical methods like finite difference to get an approximate answer to a model which more closely reflects reality but has some error introduced by those numerical methods and not the model itself). When one's developing a model, they need to be careful which approach they use. In this video, I'd say you demonstrated both approaches very well with the equilibrium and perturbation approaches, and even showed how doing both might give you more insight than just using one or the other. I certainly will be rewatching this video over and over and hope you make many more like it. I'll also send it to my old professor and see what he thinks.
@oliviab6415
As a pure mathematician, differential equations as mathematical modeling tools have always been a danger zone for me. I appreciate this breakdown. Great work, and thank you!
@yongmrchen
What an elegant model! As an extension, two revisions could be made to make the model capture more realism: (1) instead of assuming all cars have the uniform length, drawing car length from a probability distribution from empirical data, and (2) instead assuming the same car velocity, draw it from a certain distribution.
@RMDragon3
Nice video!! Personally, I prefer the quote "All models are wrong, some models are useful", meaning that no model represents the real world perfectly, but some models can allow us to understand how something works
@krillion177
What an incredible video, earned a subscriber. I love how you're able to take difficult sounding concepts and turn them into easy, understandable material, as well as giving some good general tips on the practice of mathematical modelling. Not once in the video was I bored, disinterested, or distracted! As for critiques, I will say that the audio occasionally peaks which makes the sound quality noticeably less smooth, also there's quite a lack of audio other than your voice, adding some extra music in the background or something would make it feel a lot more lively.
@Celastrous
Amazing video! I used to love messing with differential delay models during my EE degree. I never did traffic, but I did some more curiosity-driven equations wrt signal analysis
@peterdavis9403
I used to do timing of packages across a fixed length scale on a conveyor line to maximize weighing time and throughput in packages per minute. Faster belt speeds got more packages through but less time on the scale yielding a sharp peak for optimal speed. I've often thought of this as being analogous to traffic flow. Also electricity is electrons moving to "holes" thus a flow of "holes". Cars can only move into holes, but speeders fill the holes and cause those amplified ripples that slow traffic or cause accidents. Worse is the speeding of traffic on on ramps that force stops at the end by not matching traffic speed and yielding the right of way to existing traffic. This causes more ramps cars to force their way in. This is a positive feedback loop short circuiting the flow on the main highway.
@davidfarning8246
Interesting timing. This past weekend I was entering a construction area where two lanes merged into one. I was able watch with horror as a great big pickup weaved through traffic as everyone else was slowing down. Luckily, when (not if) the truck crashed, they swerved to avoid a cement barrier and hit the guard rail. They caused harm to no one but their vehicle. I spent the rest of the drive thinking about how cars might communicate with each other to eliminate or reduce this sort of behavior. There is the potential for a lot of interesting communication methods... but for they most part they depend on not having any 'bad actors' in the mix.
@federicobalzarotti2539
Hello! Thanks for all your input! I solved it but I had to change something in the final equation (26:31) because it did not reproduce the trivial solution. If you impose the disturbance z1(t)=0, the solution zi(t)=0 (or vi(t)=ve) is not achieved with the proposed equation. I think the "-v" term should be removed and rho_max is actually rho_equilibrium =(1/(L+d)).
@Beatsbasteln
hey. this video was really fascinating. i'm an audio person tho so i would like to tell you about compressors. they reduce dynamics in a signal without distorting it too much. cause atm you are basically clipping to prevent some parts of your recording to be too quiet but that creates lots of distortion and distracts audio people like me a bit. edit: btw i'd like to suggest using a lowpass filter to simulate the delayed response time of human reactions. that is usually a more natural way to soften a signal than just delaying it completely, because it also makes the transition between reacting and not reacting a bit softer
@nickadams2361
Sometimes doing math in traffic is the only way I stay sane also
@nickadams2361
great analysis, that was a fun watch. I like these sort of videos where we are doing some modeling of a very arbitrary ordinary and typical problem.
@Konchok_Dawa
This is exciting! I've thought about this exact problem before when commuting to work, and even sat down with a notebook one time to try and figure it out, nice to see my idea actually fleshed out
@allanjmcpherson
I've heard it said that all models are wrong, but some models are useful. I did a couple of classes on computational modelling for physics, so this felt very familiar. When you got to the table at the end, I found myself saying, "Well, this is just Euler's method" just before you said it. Clearly I at least remembered something! (Though given the oscillatory behaviour, this might lend itself better to something like a leapfrog.)
@RobinHillyard
Thank you. I’ve been wondering about this for a long time but never taken the trouble to model it.
@OLAFBONDD
So you confirm my ideas how every driver could make less traffic problems: reduce reaction time (control situation, react proactively) and at the same time avoid big accelerations, which result into oscillations and crashes.
@JimmieChoi93