# 3D perfectly elastic collision between two points

There is a high probability, I think, that this question is a duplicate of some other question ... but to may knowledge, it hasn't been posed in this exact manner:

Assume we have 2 points, $$P_1$$ and $$P_2$$, of mass $$m_1$$ and $$m_2$$ in a world coordinate system $$(O, \vec{i}_0, \vec{j}_0, \vec{k}_0)$$. The point $$P_1$$ is moving with the constant velocity $$\begin{bmatrix} v_{x1i}\\ v_{y1i}\\ v_{z1i}\end{bmatrix}$$ while the point $$P_2$$ is stationary. The point $$P_1$$ undergoes a perfectly elastic collision with $$P_2$$. How will these two points move after the collision?

## My attempt

This problem is about the conservation of linear momentum: therefore the momentum of the system formed by these two points remains constant. Before the collision the momentum of the system is: $$\vec{P}_{init} = m_1\cdot \begin{bmatrix} v_{x1i}\\ v_{y1i}\\ v_{z1i}\end{bmatrix} + m_2 \cdot \begin{bmatrix} 0\\0\\0\end{bmatrix}$$

After the collision, the linear momentum of the system is: $$\vec{P}_{fin} = m_1\cdot \begin{bmatrix} v_{x1f}\\ v_{y1f}\\ v_{z1f}\end{bmatrix} + m_2 \cdot \begin{bmatrix} v_{x2f}\\v_{y2f}\\v_{z2f}\end{bmatrix}$$ The unknows are $$v_{x1f}, v_{y1f}, v_{z1f}, v_{x2f}, v_{y2f}, v_{z2f}$$. But we have only three equations $$\vec{P}_{init} = \vec{P}_{fin}$$ and six unknowns ... One can also use the law of conservation of energy to obtain another equation: $$\frac{m_1}{2} \cdot \left(v_{x1i}^2 + v_{y1i}^2 + v_{z1i}^2\right) = \frac{m_1}{2}\cdot \left( v_{x1f}^2+v_{y1f}^2+v_{z1f}^2\right) + \frac{m_2}{2}\cdot \left( v_{x2f}^2+v_{y2f}^2+v_{z2f}^2\right)$$ but there are still only four equations and six variables ...

• Largely a duplicate of this: physics.stackexchange.com/q/453393 I think Jan 21, 2019 at 13:45
• If you know the exact direction in which the balls will move after the collision, assume that to be the x-axis. Jan 21, 2019 at 15:31
• Your problem will then be reduced to solving two variables with two equations. You can then write it using the original coordinate system. Jan 21, 2019 at 15:41
• However if you really want to solve using this method, you can get some equations by using the fact that the velocities of both the bodies will be along the line of contact. All the components perpendicular to this will be zero. Jan 21, 2019 at 15:45
• Do you really mean "points"? What is the point to use 3D for this case? Why would they go off the initial line of collision?
– nasu
Mar 11, 2022 at 19:16