10/23/2007 9:45 AM
Computer Problem CP0701
Projectile Motion by the Path Integral Method
Consider a baseball of mass 200 g, projected vertically upwards. It starts at height y = 0 and finishes at height y = 4.0 m, 4 seconds later. We want to describe the motion and in particular to determine the initial velocity necessary for this motion to take place. Take G = 10.0 m/s exactly.
I would like you to determine this motion using three different numerical methods. Please use Excel, and start with the file cp0701.xls, available for download on the Ph 330 private page. (http://www.physics.sfsu.edu/~bland/courses/330/private/).
(a) On the first sheet of the workbook are set up ten equal steps of 0.4 seconds. Calculate the integral of the action numerically. Carry out a variation of the nine intermediate positions of the baseball (remember, the starting and ending positions, at t = 0 and t = 4 sec, are fixed) to minimize the action. (Leave the time values fixed, and change the values of y.) It may help to watch the graph of the trajectory as you adjust the values. Keep a record of your least value of the action, in case you can't resist the temptation to go back and change these y values after you do part (b) or part (c).
(b) Now move over a couple of columns and calculate the position as a function of time using finite differences. You have to guess an initial velocity and vary it until the height at t = 4 seconds is right. Use trapezoidal integration - that is, use the average velocity over the interval in calculating the position.
(c) Finally, use the standard wisdom about projectiles,
,
to calculate the initial velocity. Then calculate y for the series of time values given.
Summarize your results and discuss.