1. Conservation of Momentum

Consider the system comprising the spaceship and the cloud particles it sweeps up. There are no external forces acting on this combined system (engine is off).
Initial Mass: $m$
Initial Velocity: $u$
When the spaceship has travelled a distance $x$ into the cloud, it has accumulated a mass of cloud particles equal to $\rho S x$.

Current Mass $M(x) = m + \rho S x$
Current Velocity $v(x)$

$$P_{initial} = P_{final}$$ $$m u = (m + \rho S x) v(x)$$ $$v(x) = \frac{mu}{m + \rho S x}$$

2. Time Calculation

We need the time $t$ to traverse the length $l$. $$v = \frac{dx}{dt} \Rightarrow dt = \frac{dx}{v}$$ Substitute $v(x)$: $$dt = \frac{m + \rho S x}{mu} dx$$

Integrate from $x=0$ to $x=l$:

$$t = \int_{0}^{l} \frac{m + \rho S x}{mu} dx$$ $$t = \frac{1}{mu} \left[ mx + \frac{\rho S x^2}{2} \right]_{0}^{l}$$ $$t = \frac{1}{mu} \left( ml + \frac{\rho S l^2}{2} \right)$$ $$t = \frac{l}{u} + \frac{\rho S l^2}{2mu}$$ Factor out $l/u$: $$t = \frac{l}{u} \left( 1 + \frac{\rho S l}{2m} \right)$$