= Avalanche front with linear mass entrainment
{title2=$\ddot x\longrightarrow g\sin\alpha/3$}
An idealized front at distance $x>0$ down a uniform slope entrains stationary material so that its moving mass is proportional to $x$. Its <linear momentum> balance is $d(x\dot x)/dt=Ax$, where $A=g\sin\alpha>0$. Multiplying by $x\dot x$ gives $x^2\dot x^2=2Ax^3/3+c$. Every forward-moving branch tends to $\dot x=\sqrt{2Ax/3}$ as $x\to\infty$, and $\ddot x=A/3-c/x^3\to A/3$. Acceleration is smaller than the constant-mass downslope value because some momentum accelerates newly entrained material.
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