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March 1st, 2008, 05:56 PM   #1
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I have a trig/calculus/geometry problem here...

I am making a pool (billiards) simulation program and am having some problems...

I need to reverse engineer a relatively simple formula.
Imagine two spheres on a plane, a White sphere and a Black one.
The Black sphere always begins stationary.
The White one always begins traveling exactly along the X axis to the right.
The exact speed of travel is irrelevant to this problem.
Eventually the balls collide.
A portion of the momentum of the white ball is transferred to the black one.
This results in each ball proceeding off at a particular angle after impact.
I know how to calculate these angles platonicly (in the ideal sense)
However the formula that I am trying to reverse engineer, applies some kind of "fudge".
This is to say, that the momentum is transferred imperfectly in some way.
Particularly, you would expect the balls to depart each other at a right angle after impact.
However, I find that the angle they depart from one another at is less and less the closer and closer they get to a direct collision.
The more "oblique" the angle of impact, the closer the departing angle is to 90 degrees.
I need to be able predict the resulting angle of each ball.
What formula can I use that will mimic the following data as closely as possible?

Start = X,Y coordinate of ball.
TRAV = A portion of their travel AFTER impact.
TRAV IS NOT THE TOTAL DISTANCE TRAVELED AFTER IMPACT.
IT IS ONLY USED TO DETERMINE THE ANGLE
Angle = The angle of travel after impact (in radians)

White Start 129,130
Black Start 174,147
TRAV WHITE 182,126 462,40 Diff=280,-86 Angle=-0.29799
TRAV BLACK 175,151 203,231 Diff=28,80 Angle= 1.23412

White Start 129,130
Black Start 184,146
TRAV WHITE 192,123 370,40 Diff=178,-83 Angle=-0.43631
TRAV BLACK 188,154 230,234 Diff=42,80 Angle= 1.0873

White Start 129,130
Black Start 174,144
TRAV WHITE 175,121 287,38 Diff=112,-83 Angle=-0.63776
TRAV BLACK 182,153 250,235 Diff=68,82 Angle= 0.87846

White Start 129,130
Black Start 184,141
TRAV WHITE 175,124 259,38 Diff=84,-86 Angle=-0.7971
TRAV BLACK 192,147 302,234 Diff=110,87 Angle= 0.66917

White Start 129,130
Black Start 184,138
TRAV WHITE 171,125 234,37 Diff=63,-88 Angle=-0.94947
TRAV BLACK 196,144 378,234 Diff=182,90 Angle= 0.45924

White Start 129,130
Black Start 184,135
TRAV WHITE 168,126 212,54 Diff=44,-72 Angle=-1.03242
TRAV BLACK 199,139 461,217 Diff=262,78 Angle= 0.28935

White Start 129,130
Black Start 184,132
TRAV WHITE 167,128 207,75 Diff=40,-53 Angle=-0.92428
TRAV BLACK 200,134 378,158 Diff=178,24 Angle= 0.13402
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