Suppose two billiard balls of same mass $m$ moving in the same axis $x$ with the same velocity and at some time $t=0$ they collide. No other forces are acting on the two billiard balls. Therefore the total momentum before the collision must be equal to the total momentum force after the collision, that is:$$\overrightarrow{P}_i=\overrightarrow{P_f}=0$$ But we can have zero total momentum even if the two balls move in opposite directions in $y$ axis with the same velocity after the collision. Both momentum and energy are then conserved. So, how one can predict the direction where the balls will move after the collision? (If the direction of force during collision is known then the direction can be predicted. But there isn't a general rule for the direction of force when two objects collide).
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$\begingroup$ The force during the collision will pass through the point of contact normally. $\endgroup$– SmarthBansalCommented Feb 13, 2020 at 17:11
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$\begingroup$ Is there friction involved here? Are these real balls, or just point masses? $\endgroup$– John AlexiouCommented Feb 13, 2020 at 18:49
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$\begingroup$ Read this post for a visualization of the momentum exchange when motion isn't collinear. $\endgroup$– John AlexiouCommented Feb 13, 2020 at 19:45
2 Answers
In the ideal case the forces act along the line joining the centres of the two balls.
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$\begingroup$ Why in ideal case this happens? Is this because no friction is assumed? $\endgroup$ Commented Feb 13, 2020 at 17:31
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$\begingroup$ The only contact forces are normal forces at the point of contact - no frictional (non normal) forces act. $\endgroup$– FarcherCommented Feb 13, 2020 at 17:35
The scenario you present (balls not rebounding along the previous direction) is actually what usually happens in real life. the direct rebounding only happens if the initial velocities are co-linear. In the case of one ball at rest, then the velocity vector of the moving ball would need to be directed toward the center of the resting ball.
If the velocity vectors are not co-linear, then the balls will not have a direct impact, and the post-collision directions will be rotated from the original. to predict the directions you would need to know the offset of the velocities or the angle between the center-to-center line and the velocities.
And since you're talking about billiards, the spin ("English") of each ball will affect the post-collision directions, too.
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$\begingroup$ But in my scenario momentum is zero before and after the collision. In which axis is zero it doesn't matter. $\endgroup$ Commented Feb 14, 2020 at 11:27
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$\begingroup$ You asked about the directions after the collision. I said that it depends on how the velocity vectors align with the centers of mass connector line. It's possible for the post-collision directions to be along lines different from pre-collision. I agree that the axes don't matter, but the offset of velocities does matter. $\endgroup$– Bill NCommented Feb 14, 2020 at 16:15