Non-trivial Projectile Problem

Hey everybody — I’ve been busily revising and expanding the first edition of my computer algebra handbooks “wxMaxima for Calculus I” and “wxMaxima for Calculus II”, and I’m enjoying a lot of support from the Maxima community (thanks!). I’m planning to release the next edition in Summer 2016 to address errata, add a code glossary, expand the exercise sets by about 30% and slightly expand the topics covered in the text. In particular, I’ve started to spend a lot of time finding programming solutions to mechanics problems that have no closed-form solution, so Edition 1.1 will be a little more complete on the programming end, in both the Examples and the Exercises.

I spent my afternoon today exploring a projectile launch at 5 m/s from a height of 3 m landing at a height of 0 m. My main interest was to nail down the maximum range and the launch angle corresponding to the maximum range, and this problem actually requires a numerical solution. *find_root* was able to solve the problem, but I also wrote a do-loop to find the maximum range by trial and error — launching the projectile from 10,000 different angles from -90 deg. to +90 deg. and keeping the largest range at each trial using an if-then statement. It strikes me as an excellent lab activity for an introductory mechanics lab for engineering/science students.

Today’s project will likely become the first in a series of monthly articles that I will begin publishing next summer — complete with all the polish you’d expect of a decent blog. For now I’ll continue working on my final revisions to the books, and I’ll just post an attractive visualization of the numerical method for your amusement:

maxout_5632_7

Thanks for your support!

Zak

The plucked string.

(Edit:  the wxMaxima calculus books are done)

 

Today I used a Fourier expansion to animate the behavior of a plucked string subject to frequency-dependent damping.  The pluck is modeled by a sharp triangular pulse in the initial state.  This is a good way to visualize what happens on a guitar string:  the motion of the string is the superposition of many standing waves of different frequencies, but the higher frequencies die out faster than the lower ones.  Eventually all that remains is the fundamental mode (the lowest note sounded by the string).  I plan to write up a proper article in the next month to address the mathematics and the CAS coding behind the animation.  Enjoy the movie!

maxout_12

wxMaxima for Calculus I and Calculus II

I’ve been on sabbatical for several months to write a series of books aimed at teaching computer algebra (using the open-source CAS wxMaxima) to lower division engineering, physical science and applied mathematics majors.  The books are designed primarily to work as 1-semester 1-unit lab manuals (presumably used concurrently with the corresponding math course), but they will also provide a valuable “by example” reference for students studying computer algebra independently.  This June I will publish the first two books as free .pdf files:  “wxMaxima for Calculus I”, and “wxMaxima for Calculus II” under a CC-BY-NC-SA license allowing anyone to modify and/or redistribute the work for non-commercial purposes.  This site will host the original .tex files and image repositories for those who want to create derivative works.  I will also post an affordable print-on-demand link this summer.

I plan to write four additional books in the coming years for Pre-Calculus, Multivariable Calculus, Linear Algebra and Differential Equations (maybe that’s the next sabbatical project?).

Here’s some CAS candy to keep you entertained in the meantime:

lissajous

Stay tuned,

Zak Hannan

Instructor of Mathematics and Physics

Solano Community College, Fairfield, CA