September 21, 2006 --- Class 8 --- Printing From Nations to Lindley,
	A Body Falling in a Position Dependent Potential (continued), 
		Graphing the velocity ratio 

Activities:

	Printing a Nations file in Lindley

		The system manager finally told me how to print from nations to
		Lindley.  On Nations, use this command:

		lp -d bl-uits-porkpie.ads.iu.edu:lh030prt  file.ps

		The file should be a PostScript file.  I tried a plain text
		file and it did not work.  To convert a text file to
		PostScript, use the enscript command

		enscript -o file.ps file.txt

		Then you can print the Postscript file as above.  I set up
		an alias and put it in my .alias file so I don't need 
		to remember the long lp command:

		alias lh030     'lp -d bl-uits-porkpie.ads.iu.edu:lh030prt'

		Now, I just need to type:

		lh030 file.ps
		
		and the file is printed in the Lindley 030 Apple cluster.


	Discussion of an object falling in a position dependent potential

		In our last class we discussed in detail the adaptive step
		size algorithm and code adapt_step.c to integrate the 
		equations of motion for a falling body in a position 
		dependent potential.

		We ran the code in adapt_step.c to see how it increases
		or decreases the step size depending on how accurately our two
		estimates of the velocity agree with each other.  
		With different initial step sizes and print times, we 
		found that the step size could increase or decrease depending 
		up whether we picked too large or too small an initial step 
		size.  Sometimes the print time prevented the step size
		from increasing because there would not a integer number
		of steps between print times.

	Extrapolation back to height = 0

		Since the integration goes to the next print time and we stop
		when the height is negative, we need to find the velocity when
		the height is zero.  It is easy to do this neglecting the
		acceleration.  However, if the print time is large, this may
		become a dominant source of error.  We examined this by testing
		different print times.

		A second code, to extrapolate back to height = 0, using uniform
		acceleration was discussed.  The
		code is in ~sg/adapt_step_nonlin_interp.c.  We compiled the
		code and ran it to verify that it improves the calculation of
		the velocity at height = 0.

	Finding the ratio to the naive velocity

		The problem is stated in terms of the ratio of the velocity in
		the position dependent potential as compared with constant
		acceleration g.  Of course, we can find the velocity with
		constant acceleration, so the code currently prints that out
		on a line by itself after the velocity from integration with
		the position dependent acceleration is printed.  We may use
		AWK to get the ratio of velocities.

		We use the awk file ~/sg/vel_ratio.a.  It contains two lines:
/^V/{vel=-$4;  rec=NR}
NR==rec+1 && NR >1  { ratio =vel/$1; print ratio} 

		We next ran the adapt_step program with several guesses for the
		initial height and found that as we increase the height the
		ratio of velocity in a position dependent potential to velocity
		assuming constant acceleration g is decreasing.  It is a bit
		boring to keep guessing values that will get us to 0.99 as the
		ratio, so we realized that we are really trying to find the
		root of an equation:

		ratio( height) = 0.99,
		and we need to find the value of height that solves this
		equation.

	Graphing the velocity ratio as function of the height via shell script

		It is relatively easy to make a graph of the velocity ratio
		as a function of the initial height.  We can use a foreach
		loop to set the initial height.  Then, we need to manipulate 
		the output of adapt_step to get the velocity ratio.  This
		last step is done with an awk script.  The
		awk file ~sg/vel_ratio.a  contains two lines:

/^V/{vel=-$4;  rec=NR}
NR==rec+1 && NR >1  { ratio =vel/$1; print ratio}

		Below is a shell script to run adapt_step and pipe the output
		through awk.  If you run this at the screen, you will
		find all the prompts from adapt_step, which are written
		to the standard error, mixed up with the output from the
		awk script, which are written to the standard output.
		If you redirect the standard output to a file with the ">"
		symbol, only the output you want will wind up in the file.

#!/bin/csh

	foreach height ( 20000 40000 60000 80000 100000 120000 140000 160000)
		echo -n $height "  "
		adapt_step <<EOF |awk -f ~sg/vel_ratio.a
		$height
		.001
		.1
EOF
	end


	Root Finding

		We spent the last few minutes of class discussing how to
		find the root of an equation.  This would let us solve for
		the height.  In our next class, we will study this more
		carefully.