Decelerating Force of Dust on Spaceship - Conservation of Momentum

[flamespirit919]

Homework Statement

A spaceship of frontal area 25 m² passes through a cloud of interstellar dust at a speed of 1.0x10⁶ m/s. The density of dust is 2.0x10^-18 kg/m³. If all the particles of dust that impact on the spaceship stick to it, find the average decelerating force that the impact of the dust exerts on the spaceship.

Homework Equations

$p = mv$
$\frac{\text{d}p}{\text{d}t}=F$

The Attempt at a Solution

I tried using the formula for conservation of momentum $$m_1v_1+m_2v_2=m_1v^{'}_1+m_2v^{'}_2$$ and got to $$m_1v_1=\left(m_1+m_2\right)v_2$$ But I got stuck here and realized I couldn't solve for the final velocity without knowing the mass of the spaceship.
So I tried using the rate of change of momentum by using $$m_1a_1+m_2a_2=0$$I assumed the initial acceleration of both masses was 0 and it made sense that the final acceleration of the spaceship and the dust would cancel out. Again, I couldn't figure out how to solve anything without knowing the mass of the spaceship. For each of these I was taking the mass of the dust to be $25\cdot2.0\times10^{-18}~kg$

I know the answer is supposed to be $5\times10^{-11}~kg/s$ or $5\times10^{-5}~N$

Some guidance in the right direction would be much appreciated. Thanks!

 

[mfb]

But I got stuck here and realized I couldn't solve for the final velocity without knowing the mass of the spaceship.

You can keep it as variable, it will cancel once you consider the limit for small time differences.

An alternative approach: Ignore the motion of the spacecraft . The front area and dust density tell you how much dust it will accumulate per time, and the speed tells you how much the dust has to be accelerated. Multiply both and you are done.

 

[flamespirit919]

So the spaceship will accumulate dust at $5\times10^{-11}~kg/s$, but I'm still not quite sure how to figure out the acceleration. Do I use one of the kinematic equations?

 

[PeroK]

So the spaceship will accumulate dust at $5\times10^{-11}~kg/s$, but I'm still not quite sure how to figure out the acceleration. Do I use one of the kinematic equations?

You are not asked to calculate the acceleration. You are asked to calculate the force. Which is why you don't need the mass of the ship.

 

[mfb]

It is like a rocket in reverse.
Imagine what happens within t=1s, for example. You have a mass of 5*10^-11 kg, and you want to change its speed by 10⁶ m/s. How much does its momentum change? What is the force required to do so?
Does the force depend on the arbitrary choice of one second as time frame?

 

[flamespirit919]

It is like a rocket in reverse.
Imagine what happens within t=1s, for example. You have a mass of 5*10^-11 kg, and you want to change its speed by 10⁶ m/s. How much does its momentum change? What is the force required to do so?
Does the force depend on the arbitrary choice of one second as time frame?

So the force will be 5*10^-5 N no matter the time frame, right?

Thank you so much. It makes much more sense now.

 

[PeroK]

So the force will be 5*10^-5 N no matter the time frame, right?

That is the force when the ship has its initial speed. As the ship decelerates the force will reduce. So, that equation is only valid for an instant. To work out how fast the ship decelerates over time, you need the mass of the ship and then that is a different problem.

 

[mfb]

The ship could use thrusters to keep its speed.
Within the scope of this problem the spacecraft has a constant speed.

 

[PeroK]

The ship could use thrusters to keep its speed.
Within the scope of this problem the spacecraft has a constant speed.

It could gather the dust particles and fire them out the back!