= Velocity-circle construction for elastic scattering
{title2=$\theta_{\max}=\arcsin(m/M)\quad(M>m)$}
In a nonrelativistic <elastic collision> of incident <mass> $M$ at <speed> $u$ with initially stationary <mass> $m$, the <centre-of-momentum frame> moves at $V=Mu/(M+m)$. The projectile's outgoing <velocity> in that frame has fixed magnitude $w=mu/(M+m)$. Its laboratory velocity tips lie on a <circle> of radius $w$ centred at $V\hat{\mathbf x}$. For $M>m$, a tangent from the origin gives the largest laboratory <scattering angle>, $\sin\theta_{\max}=w/V=m/M$. This geometrical result follows from <Galilean velocity addition> and <conservation of energy>.
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