= Solution
For a continuous size distribution, let $c_0(w)\,dw$ be initial <particle volume fraction> in a settling-speed interval. The <settling exposure for a particle size distribution> gives
$$
c(w,t)=c_0(w)e^{-wJ(t)},\qquad
g'(J)=\int_0^\infty b(w)c_0(w)e^{-wJ}\,dw.
$$
The spectrum continually shifts toward slower-settling particles. The deposit becomes progressively finer outward, with a smooth range of depositional distances instead of two distinct size components. Size-resolved deposition is $\rho_p(w)w\int_{t_a(r)}^\infty c_0(w)e^{-wJ(t)}dt$ per speed interval.
There is no universal monodisperse runout formula with an arbitrarily chosen mean settling speed. The box relation is
$$
R^4-R_0^4=4\mathsf F(\mathcal V/\pi)^{3/2}\int_0^J\sqrt{g'(s)}\,ds.
$$
A spectrum bounded away from zero settling speed has a finite limiting radius, but an arbitrarily slow-settling tail can extend the reach substantially or remove a finite runout bound. For example, if $b(w)c_0(w)\sim Cw^p$ near $w=0$, $p>-1$, its <Laplace transform> behaves as $g'(J)\sim C\Gamma(p+1)J^{-(p+1)}$. The runout <integral> converges only for $p>1$. A positive population with exactly zero settling speed retains <buoyancy> indefinitely and spreads without a finite radius limit in this no-entrainment model. Thus the fine tail of the actual distribution, and possible lofting when thermal <buoyancy> is restored, control the distal outcome.
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