Interesting tricky physics problem

Question

A spacecraft (with velocity v) moves towards a cloud of cosmic dust of density p₂. The dust that hit the spacecraft, sticks to it. Then the engine is turned on, so that the velocity of the spacecraft does not decrease. The cross-sectional area of the spacecraft is S₂, the cross-sectional area of the exhaustion pipe is S₁. The gas from S₁ are emitted with velocity u relative to the spacecraft. Find the density of the emitted gas p₁.

I have no idea how to solve this problem, so any help is appreciated.

Answer 1

This is an exercise in conservation of momentum.

The dust cloud, the spacecraft, and its fuel form a closed system --

I believe we are supposed to assume that there is no external force

(e.g., gravity of a nearby planet).

Therefore the combined momentum of all three components remains the same.

I will use the dust cloud as reference point for velocities.

I will chose the direction of the spacecraft's flight as positive,

and the opposite direction (of the exhaust gas) as negative.

And I assume that the spacecraft is moving too slowly for any relativistic effects.

Let m be the mass of the spacecraft, including all its fuel before entering the dust cloud.

At that time the spacecraft's momentum is mv.

That is the total momentum of the whole system, as the dust cloud is at rest.

After flying through the dust cloud for some time t, it covers distance vt,

it flies through volume vtS₂,

and thereby accumulates additional mass vtS₂p₂.

During that time t, the exhaust gas travels distance ut from the spacecraft,

has volume utS₁ and mass utS₁p₁.

The emitted gas has velocity -u relative to the spacecraft, hence

velocity v-u relative to the dust cloud.

At the time instant t the spaceship has new mass m + vtS₂p₂ - utS₁p₁, hence momentum

(m + vtS₂p₂ - utS₁p₁)v

The exhaust gas has momentum

utS₁p₁(v-u)

The expression for conservation of momentum is

(m + vtS₂p₂ - utS₁p₁)v + utS₁p₁(v-u) = mv

After simplifying, you can express p₁ in the form

p₁ = p₂(v/u)²(S₂/S₁)

You can understand this formula like this.

First consider the case u = v, S₁ = S₂, so p₁ = p₂.

That means that the spacecraft is dropping exhaust as it flies, replacing the dust

it is picking up, leaving everything as it found it.

If you double S₁ you can cut the density p₁ in half, leaving the same mass.

If you double the speed u you also double the mass of the exhaust.

Therefore the effect on momentum is factor of 4.

Hence you can cut the density p₁ by a factor of 4.