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?


1 Answer 1


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.

  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.