VI Arnol'd Diff Eq Question about first problem.

[johnaboboiii]

Sorry guys, I am not too bright and I am trying to learn diff eq on my own (have the class next year) and I am hung up on this problem:

If the density of air at sea level is 1250g per cubic meter, at what altitude is the density half of that?

The answer is 8ln(2) but I don't know why. I'd conjecture it had something to do with the cubic meters but I really don't know. Any help would be greatly appreciated.

 

[berkeman]

No need to be sorry, johnaboboiii. Welcome to the PF, by the way. It's a great place for learning and sharing information.

BTW, homework and coursework questions like this should be posted in the Homework Help section of the PF, not in the main forums. I know that you are studying ahead on your own, so it's technically not a homework question (yet), but a question like this should still go in the Homework Help forums.

Now, what-all have you studied so far in DiffEqs? I think you can write a couple equations that will start to answer your question, and then we can work on helping you to combine and solve them.

Think of the problem as a single square-cross-section column of the air, starting at the surface of the Earth, and extending upwards to infinity. Call the cross-sectional area 1m^2 for convenience. So going upwards, each meter tick on the column will be another cubic meter of air. The bottom cubic meter of air has the highest density and the most weight, and the density of each successive cubic meter of air gets lower as you go up, right?

The pressure on the top "surface" of each cubic meter of air is just the weight of all the other cubic meters of air above it, divided by the cross-sectional area of 1m^2. The density and pressure are related by some equation -- can you figure out what that equation looks like?

 

[johnaboboiii]

pressure=k*density where k has units m^2/s^2. k is volume*gravity/area?

I'm not really sure what to do.

 

[durt]

To get started, consider a thin sliver of air a of thickness $$dH$$ at a height $$H$$ above sea-level. What forces act on this sliver? The forces pushing up have to balance those pulling down, right? See what you can do from here.

 

[johnaboboiii]

Okay, so $$PV=nRT$$ and with a density of $$1250g/m^3$$ and a molar mass of 29 ish for air I ended up with the EQ $$Pressure=m_ag/(RT)$$ and plugged in some reasonable numbers to get K~1/8. Is this a viable path? Using this constant i get an altitude of $$ln(2)/k=ln(2)/(29*9.8/(273*8.314)) = 5.54km$$ This is pretty close to the given answer but I'm concerned about whether or not this was a decent way to go about if I'm want to learn Diff Eqs.

 

[johnaboboiii]

I guess my issue is that, I could solve the differential equation i found but is there a way I should be able to figure out k that doesn't involve physics?

 

[osnarf]

Sorry to revive a dead thread, but I'm having issues with the same problem and found this on a search, already open...

Anyhow,

I get the feel from the problem that you're supposed to be able to figure it out without any information he hasn't provided or any formulas, but I don't see how you can without knowing at least something about one other point... or even the maximum height of the stratosphere, or something. You would need bounds of some sort, right? The other problems were solved in this manner.

If you look at some dH like durt said above, then you know the forces acting downward on top and upward from the bottom are equal, so the upper and lower pressures are equal, but how can you know what either one of these pressures are if you don't have some sense of what the total weight of air is, or the height of the system, and without using some ideal gas law? He even makes note of temperature, which is what made me think about that in the first place (d = p/RT = p/[constant]).