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So, for context, I am trying to analyze how this basketball machine works: https://www.youtube.com/watch?v=FycDx69px8U

I have a basic understanding of how to calculate the new velocity of the ball after colliding with the backboard (assuming an elastic collision), but how would I approach finding a transform for the plane representing the backboard if I have a coordinate that the resulting trajectory should pass through?

In other words, given an incoming trajectory and a desired resulting trajectory after a collision, how would I find the plane (or normal vector representing that plane) that would create that resulting desired trajectory?

If there isn't a direct mathematical way to solve this, would I just sequence the different planes the trajectory can collide with and find one that fits a given set of parameters?

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If we assume that during the collision, the plane only exerts a force on the ball that perpendicular to the plane itself ($\vec{F} \propto \hat{n}$), then the impulse delivered $\int \vec{F} \, dt = \Delta \vec{p} = m \Delta \vec{v}$ will also perpendicular to the plane. Thus, for a given $\vec{v}_f $ and $\vec{v}_i$, the normal to the plane $\hat{n}$ must point in the same direction as $\vec{v}_f - \vec{v}_i$.

Note, however, that it is possible for a surface to exert a force on the ball that is not strictly perpendicular to its surface. This happens, for example, when we throw a ball with a fair amount of "spin". I suspect this can be precluded by assuming that the ball does not have significant angular momentum, but I don't see an easy proof of this.

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  • $\begingroup$ Alright, so I would solve for the plane as follows? $\vec{n} = \frac{(\vec{v}_f - \vec{v}_i)}{(\lVert(\vec{v}_f - \vec{v}_i)\rVert)}$ Where $\vec{n} $ is the normal of the plane. If so, is there an equation to transform the plane to match that normal? $\endgroup$ Apr 27 at 19:30
  • $\begingroup$ @BrandonMichelsen: Yes, if you want $\hat{n}$ to be a unit normal vector. As far as how to "transform the plane", I'm not entirely sure what you mean by that, but it sounds like more of a question for Mathematics than for this forum. $\endgroup$ Apr 27 at 19:50
  • $\begingroup$ Got it, I'll ask about the transform there. Also, just to clarify, the method we discussed for finding $\hat{n}$ given $\vec{v}_f$ and $\vec{v}_i$ will work for any trajectory defined by $\vec{v}_i$, correct? In other words, it will work for any angle the trajectory approaches from? $\endgroup$ Apr 27 at 21:25
  • $\begingroup$ @BrandonMichelsen: the only time the formula won't work is when the two vectors are the same. But then you don't need to deflect the ball anyway. $\endgroup$ Apr 27 at 22:37
  • $\begingroup$ Got it. Now, one last question. If I don't know $\vec{v}_f$ but I know I want the trajectory defined by $\vec{v}_f$ to pass through a given coordinate and I have a starting coordinate, how would I find $\vec{v}_f$? $\endgroup$ Apr 30 at 6:57

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