Rocket momentum through a cloud of particles

[EmanueleFWM]

Homework Statement

A cylindrical rocket of diameter 2R and mass M is coasting through empty space with speed v0 when it encounters an interstellar cloud. The number density of particles in the cloud is N particles/m^3. Each particle has mass m << M, and they are initially at rest.
Assume each cloud particle bounces off the rocket elastically, and that the collisions are so frequent they can be treated as continuous. Prove that the retarding force has the form bv^2, and determine b. Assume the front cone of the rocket subtends angle alpha = pi/2.

Homework Equations

The Attempt at a Solution

The solutions textbook is saying that every particle that hits the rocket has a momentum mv on the horizontal axis, and none after the collision since they get reflected straight up, and from there you can easily prove that the retarding force is bv^2.
Shouldn't the momentum of the particles on the x-axis before the collision be 0 though, since they are at rest, and the one of the rocket MV, and they both stay the same after the collision? The particles get deflected up and down, so they obtain momentum on the y axis, but the momentum of the particles going up cancels that of those going down, so the rocket should not lose any speed on the x-axis neither gain any on the y-axis going through the cloud.
Why is this not the case?

 

[PeroK]

Homework Statement

A cylindrical rocket of diameter 2R and mass M is coasting through empty space with speed v0 when it encounters an interstellar cloud. The number density of particles in the cloud is N particles/m^3. Each particle has mass m << M, and they are initially at rest.
Assume each cloud particle bounces off the rocket elastically, and that the collisions are so frequent they can be treated as continuous. Prove that the retarding force has the form bv^2, and determine b. Assume the front cone of the rocket subtends angle alpha = pi/2.

Homework Equations

The Attempt at a Solution

The solutions textbook is saying that every particle that hits the rocket has a momentum mv on the horizontal axis, and none after the collision since they get reflected straight up, and from there you can easily prove that the retarding force is bv^2.
Shouldn't the momentum of the particles on the x-axis before the collision be 0 though, since they are at rest, and the one of the rocket MV, and they both stay the same after the collision? The particles get deflected up and down, so they obtain momentum on the y axis, but the momentum of the particles going up cancels that of those going down, so the rocket should not lose any speed on the x-axis neither gain any on the y-axis going through the cloud.
Why is this not the case?

I think the particles bounce upwards in the rocket frame, but at an angle in their original rest frame.