numerical methods

* Unhappy with the lab procedures.
	- they do not really work. they work on paper.
	- you do know enough to measure things now though
	- from now on, three things:
		* learning basic numerical simulations (realistic)
		* characterizing a real projectile experimentally
		* combining simulation & experiment to really hit targets
		* how? USB rocket launchers
		* first: learning to calculate things
	- we will still do some regular labs, but mostly not
		
* meta-point
	- most of the time we say 'eqns with drag can't be solved analytically'
	- big deal, you all have computers in your pockets
	- why are we not using them?
	- at least learn the gist of coding
	- you will not all like it or learn it; need at least 1 per group ...
	
* lab thurs: crash course in numerical calculations (Python)
	- from now on, bring a laptop if you want
	- can use any language / prog you like
	- I'll try to teach you just enough python to get the job done
	
* after that: begin characterizing rocket launchers & working on code
	- launch velocity?
	- proper drag equation and parameters?
	- reliability / spread ? uncertainties ...
	
* this will take time: semester-long project

============

* So, how do the equations look to a computer?
	- parametric x(t), y(t) and stepping through time
		projectile: a=const or a=0 for any axis
		v(t) = vi + at
		v(0) = vi      v(dt) = vi + a*dt  an instant later
		v(t+dt) = vi + a(t+dt) = vi + at + a*dt = v(t) + a*dt
		velocity at time dt later - just add a*dt!
		do the same for position
		x(t+dt) = xi + vi(t+dt) + 0.5*a*(t+dt)^2 
					= x(t) + v(t)*dt + 0.5*a*(dt)^2
		if you know the initial conditions, you can just step through time
	
	- how to know when to stop? loops and conditions
		start with initial position, v, and accelerations
		step forward by dt
			update a, v, x  (note: can have a depend on v or x!)
		repeat, keep stepping forward
		! implement stop condition - when y=0 again, etc.
	
	- pseudocode 
		- example: projectile without drag
		- example: adding a drag force, in brief

* How do we do simple calculations with computers?
	- pseudocode
		- evaluating a function like f(x) = x^2 + 1
		- direct calculation, print result
		- function that takes parameters	
		- generating a list of values (array)
	- Python
		- declaring constants & variables (simple-math.py)
		- doing math (simple-math.py)
		- printing the result (simple-math.py)
				run and try different numbers
				repl.it search for python
				copy & paste code into interpreter:
					http://pleclair.ua.edu/ph125/python/tutorials/simple-math.py

* How to calculate a trajectory? [Python]
	- set up initial conditions
	- stepping through time
		- updating v's, coordinates
		- constructing a loop
		- knowing when to stop
	- defining the function
		- taking parameters
		- returning values
	- first examples
		- loops and sums 
				- sum n^2 and n^3 from 1 to 20
				- starting example (sums.py)
				- check answer with wolframalpha.com
		- 1D motion without drag (dropped object)
				(1D-v2.py) 
				- run and compare with known result for 3 heights without drag (b=0)
				- run one height with three different drag coefficients and compare
	- later: more tools, 2D motion
		- declaring a function (simple-function.py)
		- loops and sums with functions; recursion (sums2.py)
		- trajectory without drag - range, (x,y) list, time of flight
				(simple-no-drag.py)
		- sketch extension - repeating for different parameters (e.g., angles)
				(simple-drag.py)
				
* What we will do
	- USB foam rocket launchers
	- do we need drag? how to characterize it?
	- can we accurately model the rocket?