# Calculating point of impact in a 2d elastic collision

After an experiment for proving the conservation of momentum law using the following construction:

2 stages for the experiment:

In the first stage, a ball is placed in the beginning of a rail and left to fall. We mark its collision point with the plane (The head of $\Delta X$).

In the second stage, we put another ball on the end of the rail, the first back up at the top, let them collide in an angle and mark the hit points of both on the plane. We change the angle a few times and mark all hit points (like shown - the black dots in the sketch)

I thought - how can I mathematically prove that by changing the angle of collision, the collection of points where the balls land on the plane creates a circle with diameter $\vec X$? (shown by the black dots in the picture)

Balls are of the same mass, so we can derive from the conservation of momentum that:

$$\overrightarrow {\Delta r_1} + \overrightarrow {\Delta r_2}=\overrightarrow {\Delta X}$$

Now from the conservation of energy, we can know that (After the development of the equation $E_{ki}=E_{kf1}+E_{kf2}$):

$$\Delta X^2=\Delta r_1^2+\Delta r_2^2$$

And from this, we can imply that the angle between $\overrightarrow {\Delta r_1}$ and $\overrightarrow {\Delta r_2}$ is $90^{\circ}$

Now, of course, we can explain why the dots would create a circle by explaining that a $90^{\circ}$ triangle with the hypotenuse being the diameter, the vertex between the legs would be on the circle's perimeter.

I wondered, is there a more mathematical way to prove that? I thought about expressing the $x$ any $y$ of the vectors $\overrightarrow {\Delta r_1}$ and $\overrightarrow {\Delta r_2}$ using a parametric function of some variable $\theta$ between the balls at the collision point, but couldn't develop it to a point where it is in the form of an equation for a circle.

Do you have any other ideas?

• Please explain what's happening in this experiment, it's not clear form the sketch. What has this to do with elastic collisions? – leftaroundabout Oct 14 '17 at 18:53
• Alright, just a second. I'll add it to the question. – Yotam Salmon Oct 14 '17 at 18:53
• @leftaroundabout Edited. I hope it's clearer now. – Yotam Salmon Oct 14 '17 at 18:57
• It seems reasonably clear now; trouble is, the “result” you're trying to prove is just wrong. The landing points won't form a circle. Are you sure you got that right? – leftaroundabout Oct 14 '17 at 19:04
• Why won't they? I mean, can you explain why it doesn't happen? – Yotam Salmon Oct 14 '17 at 19:09