derive the ladder curve without calculus
In this video, I build intuition with limits to derive the bounding envelope of a falling ladder. I assume the ladder is sufficiently thin. I also assume knowledge of high school geometry and some trigonometry. Here we make use of algebraic manipulation, including fractional exponents.
Although the word "calculus" appears several times in my narration, this video is labeled "without calculus" in the sense that we are not doing any derivatives or integrals, nor setting up a difference quotient, as such--though a case can be made that we are doing calculus in disguise. Either way, the point is to help build intuition and confidence in the notion of limits, and negligible quantities, and to find the envelope equation.
I chose to use simple trig definitions repeatedly rather than a more elegant method. For example, the perpendicular lengths could be avoided by applying the Law of Cosines. The alternative and elegant approach to this problem, involves derivatives of parametric curves, while setting the ladder to unit length. I suppose I could have chosen to label (x_ell,y_ell) simply as (x,y) but here we are.
I recommend also watching my string art envelope video linked here:
• the real string art sy...
Special thanks to Patrick Sandiland for assistance with drawing the hybrid curve. Thank you JoAnne McKenny for background and Scott Seward and Family for help with audio.
Intro music was written and performed by me, Sobermath. All rights reserved.
Пікірлер: 19
its crazy how you only have 47 subs i thought it was a way more popular channel
@sobermath
5 күн бұрын
Thank you for the complement! This channel is only a year old and I've put all my effort into video content rather than marketing.
@kwiky5643
4 күн бұрын
@@sobermathgigachad !
@samueldeandrade8535
Күн бұрын
Good channels, in some sense, doesn't have subs.
Incredible !
nice, maybe you could try the catenary next? although that would be hard without calculus
@sobermath
6 күн бұрын
Thank you. Good idea! I'm actually working on such a video, for release in a few months. Stay tuned!
OMG!, You are Awesome!!!!!!!
I love your content❤
@sobermath
6 күн бұрын
i feel honored! Thank you for watching.
Damn, its crazy, as a algebra type mathematician I amazed by this video!) How you get
@sobermath
4 күн бұрын
Thank you for watching! I can't explain the fewer than 100 subscriber count previously, except that this is the first video: that I premiered, spoke on camera and ended on "to be continued." Feel free to comment on my other videos!
@SobTim-eu3xu
4 күн бұрын
@@sobermath of course!)
Thank you for the video!
@sobermath
3 күн бұрын
Glad it was helpful!
Cooooool!!!
cant you do this though a lerp or a spline? surely their are simpler way was to determine a line moving thorough 2d space
@sobermath
6 күн бұрын
Yes, there are more elegant methods. The way I was made aware of with calculus was solving parametrically and with a ladder of unit length. Much simpler!
University? I learned this in grade 11