# Classical Mechanics: Retarding force on a satellite

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1. Nov 17, 2016

### Niall Kennedy

1. The problem statement, all variables and given/known data
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

2. Relevant 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.

3. 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.

2. Nov 17, 2016

### PeroK

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

3. Nov 17, 2016

### Niall Kennedy

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

4. Nov 17, 2016

### PeroK

Do you know $M$?

5. Nov 17, 2016

### Niall Kennedy

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

6. Nov 17, 2016

### PeroK

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?

7. Nov 17, 2016

### Niall Kennedy

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?

8. Nov 17, 2016

### PeroK

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?

9. Nov 17, 2016

### Niall Kennedy

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