Collisions in Two Dimensions — Question 6

Question 6

A measured component leaves two mirror-image scattering solutions

Identical pucks collide elastically, with one initially at $(8,0)\,\mathrm{m/s}$ and the other at rest. The first puck's final $x$ component is $3\,\mathrm{m/s}$.

1. Find every possible final velocity pair.

2. Show that the first puck's velocity endpoint lies on a circle and explain the remaining ambiguity.

Question 6 - Solution

A measured component leaves two mirror-image scattering solutions

(a) Derivation and calculation

Write $\vec v_1=(3,b)$; momentum gives $\vec v_2=(5,-b)$. Elastic energy requires $$3^2+b^2+5^2+b^2=8^2\Rightarrow2b^2=30.$$ $$\boxed{\vec v_1=(3,\pm\sqrt{15}),\quad\vec v_2=(5,\mp\sqrt{15})\ \mathrm{m/s}}.$$

(b) Result and interpretation

For a general endpoint $(a,b)$, $$a^2+b^2+(8-a)^2+b^2=64\Rightarrow a^2+b^2=8a.$$ $$\boxed{(a-4)^2+b^2=16}.$$ The line $a=3$ intersects this circle twice. The two outcomes are reflections across the incident axis; a transverse-direction measurement selects one.

Physical check

The two signs cancel exactly in total transverse momentum.