Star moving through a cloud of particles

[fluxions]

Homework Statement

A star of mass M and radius R is moving with velocity v through cloud of particles of density $\rho$. If all the particles that collide with the star are trapped by it, show that the mass of the star will increase at a rate
$$\frac{dM}{dt} = \pi \rho v \left(R^2 + \frac{2GMR}{v^2}\right).$$

The Attempt at a Solution

Assuming the motion of the star is rectilinear, it's evident that in a time $\Delta t$ the star will collide with all the particles in a cylinder of length $v \delta t$, which has mass $\pi R^2 \pho v \Delta t$. This gives the first term in the desired equation after division by $v \Delta t$.

As for the second term... I'm stuck. Here's what I've tried: Consider the particles in a 'slab' (of thickness dr) of the aforementioned cylinder a distance r away from the center of the star at the beginning of the time interval $\Delta t$. This slab will experience an acceleration GM/r^2. Hence, assuming it started from rest, it will travel a distance $$\frac{GM (\Delta t)^2}{r^2}$$ in the time interval $\Delta t$. Et cetera. I don't know how to proceed from this point.

 

[tiny-tim]

hi fluxions!

(have a pi: π and a rho: ρ and a delta: ∆ )

try it from the frame of reference of the star (with a wind of particles of velocity v), and use conservation of energy