Classical Mechanics: Retarding force on a satellite

In summary, the satellite is moving through a gas with a uniform density, and is affected by a force which is inversely proportional to the square of the distance between the satellite and the particles.
  • #1
Niall Kennedy
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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.
 
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  • #2
For part a) why not focus on momentum and first consider the effect of a single particle of mass ##m##.
 
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  • #3
PeroK said:
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?
 
  • #4
Niall Kennedy said:
Something like,
Mv = (M + m)u ?
Or am I taking it the wrong way?

Do you know ##M##?
 
  • #5
PeroK said:
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.
 
  • #6
Niall Kennedy said:
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?
 
  • #7
PeroK said:
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?
 
  • #8
Niall Kennedy said:
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?
 
  • #9
PeroK said:
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
 

1. What is classical mechanics?

Classical mechanics is a branch of physics that deals with the motion of macroscopic objects and the forces acting on them. It is based on the laws of motion proposed by Sir Isaac Newton in the 17th century.

2. What is a retarding force?

A retarding force is a force that acts opposite to the direction of motion of an object, causing it to slow down or come to a stop. It can be caused by various factors such as air resistance, friction, or drag.

3. How does a retarding force affect a satellite's motion?

A retarding force on a satellite in orbit will cause it to lose energy and gradually spiral towards the Earth. This can lead to a decrease in the satellite's speed and ultimately result in its re-entry into the Earth's atmosphere.

4. How is the retarding force calculated for a satellite?

The retarding force on a satellite is typically calculated using the formula F = μAv^2, where μ is the coefficient of friction, A is the surface area of the satellite, and v is its velocity. However, the specific calculations may vary depending on the specific factors contributing to the retarding force.

5. Can a retarding force be beneficial for a satellite?

Yes, a retarding force can be beneficial for a satellite in certain scenarios, such as when it is used to slow down and control the satellite's descent during re-entry into the Earth's atmosphere. However, it can also be detrimental if it causes the satellite to lose too much energy and fall out of orbit prematurely.

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