= Relativistic head-on elastic collision with a stationary target
{title2=$E'=[2mM^2+E(m^2+M^2)]/(2Em+m^2+M^2)$}
In units with $c=1$, a particle of <rest mass> $M$ and total <energy> $E$ strikes a stationary target of <rest mass> $m$. For the nontrivial one-dimensional <elastic collision>, reverse the incident <momentum> in the <centre-of-mass frame> and transform back to obtain the displayed outgoing projectile <energy>. The invariant squared total <four-momentum> is $s=M^2+m^2+2mE$. If $M\gg m$ and $Em\gg M^2$, then $E'\sim M^2/(2m)$. If $m\gg M$, retaining arbitrary $E/m$ gives $E'\simeq Em/(m+2E)$; only the further fixed-$E$, infinite-target-mass limit gives $E'\to E$.
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