= Solution
Let $\mathbf a,\mathbf b$ be the outgoing <velocities>. In the nonrelativistic <elastic collision>, equal <masses> give
$$
\mathbf v=\mathbf a+\mathbf b,\qquad v^2=a^2+b^2
$$
by <conservation of momentum> and <conservation of energy>. Squaring the first relation and comparing with the second yields
$$
\boxed{\mathbf a\cdot\mathbf b=0.}
$$
If both particles move, their paths are perpendicular. If either <velocity> vanishes, its direction is undefined and the other particle carries all the incident <momentum> and <kinetic energy>. This conclusion depends on the stated $v\ll c$ approximation.
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