Speed of falling in atmosphere

[maxwiz]

How to derive a formula for speed of falling body in Earth atmosphere?
I know distance from surface, pressure at this distance, object mass, falling start point and point where I want to calculate the speed.

 

[jbriggs444]

Which falls faster? A kilogram of marbles or a kilogram of feathers?

Could there be some parameter that is missing from the list of things you know?

 

[maxwiz]

Thanks, I missed bottom surface of object and density. I've got them now.
But still in question.

 

[rcgldr]

Wiki article for relatively low altitudes where air density can be considered constant:

wiki_free_fall_with_air_resistance.htm

To take into account the decrease in density of the air versus altitude, you'll need to use differential equations and some type of numerical integration.

 

[cjl]

This is a distinctly nontrivial problem. Even if you make a couple of simplifying assumptions - such as that the atmosphere is of a constant density, the object has a known drag coefficient, and the atmosphere is incompressible (which should all apply if you know a bit about the object's aerodynamic configuration, and it's falling at less than about mach 0.3 in the lower couple thousand feet or so of the atmosphere), it becomes a differential equation of the form x'' = -g+k*x'². This is a nonlinear differential equation (since the first derivative of x has a power other than one). I don't personally know of any way to solve this other than computational methods, even though we made a number of simplifying assumptions to get it this far. I will admit though that I haven't studied differential equations in great detail, so there may be some way to solve this that I am not aware of.

If you want to include factors such as changing atmospheric density, or changing drag coefficient, then you pretty much have to integrate it numerically - there is no closed form solution to my knowledge.

Is there a specific problem you had in mind, or is this more of a general question?

 

[mikeph]

x'' = -g+k*x'2

can we not separate variables (in dv and dt) to get

dv/(-g + kv^2) = dt

The integral of the LHS should be take the form of a tanh^-1(v), which would imply v(t) is some tanh function... a plot which looks fairly plausible for a body accelerating to a terminal velocity.

 

[rcgldr]

can we not separate variables (in dv and dt) ...

Wiki article linked to above includes the math for this case.

 

[maxwiz]

...

Is there a specific problem you had in mind, or is this more of a general question?

Currently I want to calculate conditions under which free-falling human body can accelerate to supersonic speed in atmosphere.

 

[harrylin]

Currently I want to calculate conditions under which free-falling human body can accelerate to supersonic speed in atmosphere.

Just a side remark: I was also thinking about that supersonic jump, however I was wondering about the related question of how to calculate the speed of sound as a function of height. Probably the speed of sound at great height is different from that sea level, but this is often neglected. A quick search gave me that the speed of sound is different at high altitude, but only because of the lower temperature:

- http://www.sengpielaudio.com/calculator-airpressure.htm
- http://www.aerospaceweb.org/question/atmosphere/q0112.shtml

According to those, colder air has a somewhat lower speed of sound (and humidity has almost no effect).

 

[davenn]

Just a side remark: I was also thinking about that supersonic jump, however I was wondering about the related question of how to calculate the speed of sound as a function of height. Probably the speed of sound at great height is different from that sea level, but this is often neglected. A quick search gave me that the speed of sound is different at high altitude, but only because of the lower temperature:

According to those, colder air has a somewhat lower speed of sound (and humidity has almost no effect).

I would have thought lower air density at high altitude would contribute greatly to being able to fall faster ... less air... less air resistance... higher speed can be attained
less dense air (gas) has a lower sound propagation property, it only just happens to be colder up there at altitude

Dave

 

[cjl]

Currently I want to calculate conditions under which free-falling human body can accelerate to supersonic speed in atmosphere.

You would need to numerically integrate that one, since it involves falling tens of thousands of feet across a fairly substantial density gradient, and the drag coefficient will have the be velocity dependent (since the compressibility effects will be significant).

 

[cjl]

I would have thought lower air density at high altitude would contribute greatly to being able to fall faster ... less air... less air resistance... higher speed can be attained
less dense air (gas) has a lower sound propagation property, it only just happens to be colder up there at altitude

Dave

Density is actually not correlated with the speed of sound at all, assuming air to be an ideal gas (which is pretty darn close for this purpose). A lot of people think it is, since the air is less dense at altitude and the speed of sound is lower at altitude (both common knowledge), but the real influence comes from temperature. Within normal ranges of pressure and temperature, the speed of sound is equal to √γRT, in which gamma and R are both gas parameters (and therefore constant for a given gas, such as air). The only variable in this equation is the temperature, and the reason the speed of sound is lower at altitude is because it is colder.

 

[harrylin]

[..] less dense air (gas) has a lower sound propagation property, it only just happens to be colder up there at altitude[..]

That appears to be debunked in the first link, didn't you read it? And of course, what matters for the original poster is just to know the speed Mach1 as function of height, as provided in those links.

 

[davenn]

Density is actually not correlated with the speed of sound at all, assuming air to be an ideal gas (which is pretty darn close for this purpose). A lot of people think it is, since the air is less dense at altitude and the speed of sound is lower at altitude (both common knowledge), but the real influence comes from temperature. Within normal ranges of pressure and temperature, the speed of sound is equal to √γRT, in which gamma and R are both gas parameters (and therefore constant for a given gas, such as air). The only variable in this equation is the temperature, and the reason the speed of sound is lower at altitude is because it is colder.

thanks for that

it sort of goes against all I have learned that says increasing density ( of any material) increases speed of propagation. Didnt realize that a gas doesn't fall into that category

cheers
Dave

 

[cjl]

thanks for that

it sort of goes against all I have learned that says increasing density ( of any material) increases speed of propagation. Didnt realize that a gas doesn't fall into that category

cheers
Dave

Interestingly enough, in the absence of any other changes, an increase in density actually decreases the speed of propagation. There are a number of (provably equivalent) statements for the speed of sound, depending on which parameters you want. As stated above, for an ideal gas, it is equal to √(γRT). However, it is also (in general) equal to √(E/ρ), in which E is the bulk modulus (how difficult the substance is to compress), and ρ is the density. As you can see, as density increases, the speed of sound actually decreases. The reason why the speed of sound is higher in some dense media (such as metal or water) than in some low density media (like air) is because the bulk modulus of water or of steel is much, much higher than the bulk modulus of air, and this increased bulk modulus more than makes up for the increased density. Alternatively, it is also equal to √(dP/dρ) (that should be a partial derivative by the way, but I can't figure out the proper formatting).

One way that may help visualize this is to think of the medium in which sound is propagating as a string of masses connected by springs (so you have a mass, then a spring, then another mass, then another spring, etc). The bulk modulus of the material is like the strength of the springs, while the density is like the mass of each individual weight in the string. If you push on the end of it, introducing a compression wave, it is very much analogous to a sound wave propagating through a material.

Now, replace those springs (in your mind - or in the actual system - this wouldn't be that hard to build as a demonstration unit...) with weaker ones. This is like decreasing the bulk modulus of the material, making it more compressible. It's fairly intuitive that the wave will propagate slower.

Next, change the weights. Once again, it's fairly intuitive that heavier weights (representing a material with a higher density) will cause the wave to propagate slower, not faster. When visualized this way, it becomes a lot more clear that density has the opposite effect of what is commonly thought.

Unfortunately, in terms of sound speeds in real, physical materials, it's impossible to perfectly separate out bulk modulus and density like this, and this is where the confusion often arises. Air, being a gas, has an incredibly low bulk modulus, and thus a very low sound speed. Water, being a liquid, has a very high bulk modulus, and thus a higher sound speed, despite having a density approximately a factor of a thousand higher. Many people see this fact, and they immediately jump to the conclusion that density is the reason, rather than the compressibility of the material, leading to the widespread nature of the misconception.