Classical Mechanics: Retarding force on a satellite

[Niall Kennedy]

Homework Statement

A spherical satellite of radius r is moving with velocity v through a uniform tenuous atmosphere of density ρ. Find the retarding force on the satellite if each particle which strikes it (a) adheres to the surface and (b) bounces off it elastically.

I know the answer should be: -ρAv²

Homework Equations

I am not fully sure one what equations are relevant but I am thinking, for part (a) conservation of momentum and for part (b) conservation of kinetic energy.

The Attempt at a Solution

For part (a):
This is what I tried but it did not really lead to anything that makes sense, maybe I set it up wrong or took a wrong approach?
Mv + dm(v - u)= (M + dm)(v - dv)

For part (b):
I intended to use the conservation of kinetic energy but I ended up getting confused on the set up of it.

 

[PeroK]

For part a) why not focus on momentum and first consider the effect of a single particle of mass $m$.

 

[Niall Kennedy]

For part a) why not focus on momentum and first consider the effect of a single particle of mass $m$.

Something like,
Mv = (M + m)u ?
Or am I taking it the wrong way?

 

[PeroK]

Something like,
Mv = (M + m)u ?
Or am I taking it the wrong way?

Do you know $M$?

 

[Niall Kennedy]

Do you know $M$?

Sorry, I probably should have explained. I don't know what M is but I was using it as the mass of the satellite.

 

[PeroK]

Sorry, I probably should have explained. I don't know what M is but I was using it as the mass of the satellite.

Yes, I understood that. But, if you don't know $M$ and it probably isn't intended to be a factor in the answer, then you may need to think again.

Can you estimate $u$ from that equation?

 

[Niall Kennedy]

Yes, I understood that. But, if you don't know $M$ and it probably isn't intended to be a factor in the answer, then you may need to think again.

Can you estimate $u$ from that equation?

Oh okay, that makes a lot of sense.

In terms of M, yes but without M, no. So could I make the assumption that the particles in the atmosphere are at rest and say that the mass of the particles hitting the satellite = ρA which hit the satellite at -v?

 

[PeroK]

Oh okay, that makes a lot of sense.

In terms of M, yes but without M, no. So could I make the assumption that the particles in the atmosphere are at rest and say that the mass of the particles hitting the satellite = ρA which hit the satellite at -v?

Let me help you out. The idea is that if $m$ is very small compared to $M$, then you can ignore the negligible change in velocity over a short time. I'm not sure whether this has been mentioned somewhere in your course or whether you are expected to be able to think on your feet.

Actually, changing the frame of reference, so that you imagine a large satellite being bombarded by a stream of small particles is a good idea. Especially for part b).

Does that make sense?

 

[Niall Kennedy]

Let me help you out. The idea is that if $m$ is very small compared to $M$, then you can ignore the negligible change in velocity over a short time. I'm not sure whether this has been mentioned somewhere in your course or whether you are expected to be able to think on your feet.

Actually, changing the frame of reference, so that you imagine a large satellite being bombarded by a stream of small particles is a good idea. Especially for part b).

Does that make sense?

That makes sense, a lot of sense actually, thank you!

That's something I've used a lot before and should really think of straight away, I think this question has just been annoying me for too long haha