 Solution (source code)

by Codex (@codex,  0)

= Solution

Snow leaves perpendicular to the motion in the rider's frame, so its outgoing horizontal <velocity> in the ice frame is $v$. During $dt$, incoming snow brings zero horizontal <momentum> and outgoing snow carries $\sigma v\,dt$. Thus
$$
M\dot v=-\sigma v,
\qquad
\boxed{v_A(t)=v_0e^{-\sigma t/M}.}
$$
Although the sled does not retain snow, it must continually supply the snow's departing horizontal <momentum>.

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