Consider the following Diagram in which a Cue Ball (A) of mass M is shot twice at another pool ball with identical mass M.
When the force with which the cue ball (A) is hit (v1) is increased (v2) it seems that the angle which the cue ball is reflected is decreased, while the angle that the other ball is deflected is increased.
Is this thinking accurate? Or is this an illusion, and does the value X remain constant no matter what the force V?
Is there a way of calculating the exact of angle of deflection of the cue ball (x) and the angle of the deflection of the other ball (y), if you know its mass and the angle of incidence at which it hits? The second ball is resting and they have identical mass.
**Note: This is a simplified example where the ball is not spinning (Or in pool terms, "full stun."). But if you would like to provide a general formula that also accounts for spin, please do!
Best Answer
http://billiards.colostate.edu/physics/Alciatore_pool_physics_article.pdf
For a stun shot, the angle y of the object ball (OB) is along the center of the ghost cue ball to the OB. A ghost CB would be where the CB is at the moment of collision. The CB deflection angle x = 90 - y. From your picture, Y = 180 - y and X = 180 - x.
The reason for this is on a stun shot, there are no transfer of angular momentum. The force is transmitted along the line from the center of the CB to the OB, hence the reason why OB is deflected at angle y. Since all the force is transmitted along this line, there are no linear momentum of the CB along this line, so the CB will deflect at 90 degree - y.
Speed of the CB has not effect on the angle with the caveat that the speed is small enough such that the deformation of the CB & OB is negligible.
If you do see an angle difference with increased speed, that's due to throw. On the throw, the CB is rolling forward with no sliding (assuming the distance you strike with the cue is far enough for the CB to get into rolling with no sliding). At contact, the angle of deflection remains the same; eg y for OB where y is the angle between the path of motion to the line from the center of CB to OB, and x is 90 degree - y. Because the CB is rolling, after contact, there would now be rolling with sliding, which produces a contact force which causes the CB to curve along the path of a parabola until it reach rolling without sliding. This contact force is wrcosx/sqrt(v^2 + 2vwrsinx + (wr)^2), where w is the angular speed of the CB at contact, r is the radius of the CB, v is the speed of the CB, and x is the angle of deflection.
The final angle that the CB would travel -- I'm still working on that.