sprocket wrote:Good job Mike; that makes perfect sense. I stated that the perfect board would be different depending on the entry angle but I can see how that is not true if the goal is to gt the head pin to move directly towards the 7 pin. It WOULD be true at any point EXCEPT the exact point of cantact. At that point, if you took a freeze frame you would not know where the ball came from, you would just know whether or not it was striking the headpin at the perfect position.
Now all that being said, imagine the ball coming in at the shallowest angle imaginable that can still get the ball to contact the head pin at the theoretically perfect spot. That ball has very little if any momentum towards the five pin so although it should in theory clear the 1-2-4-7 and 3-6-10 perfectly, one would think it would leave a weak 8 pin. The funny thing is you seldom see just a weak 8 pin. Usually the 10 pin is with it. That doesn't make sense if the ball is contacting the head pin at the perfect spot. It tells me that perhaps the position of the ball changes instantaneously when it contacts the headpin from that shallow angle, and before the 3 pin can move directly towards the 10 pin the ball is already deflecting towards the 3 pin and moving WITH the 3 pin. So it maintains contact with the three pin for a little longer period of time, effectively pushing the 3 pin off its perfect line....and an 8-10 is the result.
By the same reasoning that the ball has to hit the head pin with a 3.33" offset to drive the head pin into the 2 pin, the ball also has to hit the 3 pin with a 3.33" offset (on the left) to drive the 3 pin into the 6 pin.
The 3 pin is offset right 6" from the center of the head pin. and 10.4 inches back. (6*sqrt(3))
The path the ball needs to take after contacting the head pin is to a point 2.67" right of the center of the head pin, and 4.62" back.
To be complete, the center of the ball at impact with the head pin would be 3.33" right (lateral direction), and 5.77" in front (linear direction) of the center of the head pin.
At the point the ball correctly impact the 3 pin, it would be 3.33" left and 5.77" in front of the center of the 3 pin.
That location in reference to the head pin is 2.67" right, 4.62" behind the head pin.
(6-3.33) = 2.67
(10.39-5.77) = 4.62
10.39 is the linear distance from the head pin to the line between the centers of the 2 and 3 pin.
sqrt(12*12-6*6) = 10.39
The ball will have to travel 0.66 left, and 10.39 back between the impact with the head pin, and the impact with the 3 pin.
atan(0.66/10.39) = 3.63 degrees
I have to refresh my physics on elastic collisions to include Coefficient of Restitution of both the ball, and the pin to determine the relationship between angle of entry, ball speed, and weight.
In addition a ball has rotational momentum that can alter the path after impact.
So a ball that hits the pocket, but at a weak angle. (take a left hander crossing over as an extreme example) will not magically deflect back to the left to be in the proper place to drive the 3 into the 6.
By watching the 1-2-4-7 pins you can tell if you are hitting the head pin correctly, then you observe the 3, 6, 10 to determine if the angle, deflection, and speed combination are correct.