September 22, 2005 --- Class 7 --- A Body Falling in a Position 
	Dependent Potential  (continued), Graphing the velocity ratio 

Activities:

	Discussion of an object falling in a position dependent potential

		In view of the length of time since we last discussed this
		problem, we reviewed what we previously discussed.

		We wish to solve problem 3.4 of CSM.  We drop a ball
		from high above the surface of the earth (neglecting air
		resistance!).  From how great a height must we drop it so
		that it's velocity when it hits the ground is 1% different
		from what it would be assuming constant acceleration.

		There are several potential sources of error:
		1) We know from the coffee cooling problem that when we do
		numerical integration there can be a step-size error.  So
		we must be careful about this.

		2) When we integrate the equations in time, we won't
		necessarily stop integrating exactly when the object
		reaches height 0.  We may have to interpolate to find the
		velocity when the height is zero.

		3) We also must find what height results in a 1%
		difference.  Do we guess initial heights?  How do we
		accurately find the height that corresponds to 1%?

	Step size error

		We have three approaches to the step size error issue.
		One possibility is to vary the step size and see how the
		answer varies.  The accuracy on the velocity at height=0
		must be much greater than 1%.  The accuracy will depend on
		height and step size.

		We can also try to find a better method than Euler or
		Euler-Cromer.  These are each first order methods.  The
		Euler-Richardson method is one order higher.  Try running
		this code with different step sizes.  Note the accuracy of
		the energy and the position.  (Interestingly, the
		difference is negligible for the velocity.)

		Another scheme involves using the Euler-Cromer algorithm
		with different step sizes to extrapolate to zero step size.
		This was described on the board and is coded in
		adapt_step.c.  This code will increase or decrease the step
		size to achieve a desired level of accuracy during the
		integration.

		We examined 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.  We then ran
		the code with different initial step sizes and found that the
		step size could increase or decrease depending up whether we
		picked too large or too small an initial step size.

	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