Do Hot and Cold Air Balloons Fall at the Same Rate?

[bwana]

Certainly a wooden ball and a stone fall with the same velocity. Would a balloon of cold air fall with the same velocity as a balloon of hot air?

As a corollary, why does hot air rise over cold air? If you claim that cold air is denser, then the same argument fails when you compare a wooden ball with a stone falling.

 

[DrClaude]

Objects will fall through air with the same acceleration provided that the effects of air are the same on the different objects. In the case you are thinking about, the density of the object must be much greater than the density of air to be able to neglect the effects of buoyancy.

 

[CWatters]

Certainly a wooden ball and a stone fall with the same velocity. Would a balloon of cold air fall with the same velocity as a balloon of hot air?

If both balloons had the same mass of air in them then hot one would be larger so there would be more air resistance/drag making it fall slower.

If both balloons were the same volume the hot one would have less mass in it so would be less dense. It might rise rather than fall.

If you made your "balloons" out of a rigid material then they could have the same mass and volume. In which case they would fall equally fast.

 

[CWatters]

As a corollary, why does hot air rise over cold air? If you claim that cold air is denser, then the same argument fails when you compare a wooden ball with a stone falling.

Wooden and stone balls only fall at the same speed in a vacuum. In air there is a slight difference due to buoyance but it's usually too small to be noticeable because both are much denser than air.

In water the difference is much less and the wooden ball floats and the stone sinks.

In Mercury I think both float but don't hold me to that.

 

[Nidum]

An interesting experiment that you can do is to find a lightweight thin transparent plastic drum with lid , secure your test objects inside one at a time and measure the times taken for the drum to fall from a set height .

Can you guess or better work out how the times of fall will vary ?

 

[bwana]

Thank you for your replies. Yes I thought of air resistance, buoyancy, etc. But these are all macroscopic quantities. I was trying to understand the effect of the medium through which the objects fall in a statistical mechanical point of view.

Consider the molecules of hot air and those of cold air. Now these particles are colliding with the medium that surrounds them. What is different about the collision of a 'hot molecule' with the medium vs the collision of a cold molecule that results in the different behavior of hot air vs cold air? Certainly, hot air molecules have a greater velocity, a greater kinetic energy. How does this result in these particles 'rising above' the slower particles? Can we compare the vertical descent of a dropped bullet vs a fired bullet through a container of bullets? Assuming no friction, won't these bullets hold the same altitude?

Again, recall that the argument is not about density (another macroscopic quantity). The Galilean experiment illustrates the similar behavior of differently massed particles in the absence (or negligible presence) of a surrounding medium.

 

[Herman Trivilino]

Thank you for your replies. Yes I thought of air resistance, buoyancy, etc. But these are all macroscopic quantities. I was trying to understand the effect of the medium through which the objects fall in a statistical mechanical point of view.

Macroscopically all objects fall the same in a vacuum. Microscopically all objects fall the same if they don't collide with other objects (such as the particles that make up a medium) as they fall. Certainly you can make things more complicated by finding the conditions under which two objects will fall the same in the presence of air resistance and buoyancy, but still, on a microscopic level, the effects of the collisions with the particles that make up the medium provide the microscopic explanation.

Consider the molecules of hot air and those of cold air. Now these particles are colliding with the medium that surrounds them. What is different about the collision of a 'hot molecule' with the medium vs the collision of a cold molecule that results in the different behavior of hot air vs cold air?

The notion of hot and cold is a macroscopic consideration. Microscopically you have fast-moving molecules and slow-moving molecules.

Certainly, hot air molecules have a greater velocity, a greater kinetic energy. How does this result in these particles 'rising above' the slower particles?

If you compare a collection of fast-moving molecules with a collection of otherwise identical slow-moving molecules you will find that, on average, the fast-moving molecules have more distance between them. You can get a feel for this effect by shaking a box that contains a collection of ping pong balls. If you shake the box harder you make the balls move faster and their separation distance increases. This is the microscopic explanation for the macroscopic phenomenon known as thermal expansion. Certainly the effect can be explained in greater depth with a mathematical analysis, but this is the gist of it.

 

[bwana]

Your words still do not explain why the more rapidly moving particles should ‘float’ above the slower moving particles in a gravitational field

 

[A.T.]

I was trying to understand the effect of the medium through which the objects fall in a statistical mechanical point of view.

I think you cannot just look at a single collision in isolation, and wonder what makes the faster particle go up. The probability of different types of collisions likely depends the gradient of temperature. Faster particles do not just rise individually through slower particles. They can also pass on their momentum and become slower. Also keep in mind that for temperature, the particle movement relative to the bulk movement is relevant, not relative to some frame where the pocket of hot air is moving.

 

[cjl]

If you compare a collection of fast-moving molecules with a collection of otherwise identical slow-moving molecules you will find that, on average, the fast-moving molecules have more distance between them. You can get a feel for this effect by shaking a box that contains a collection of ping pong balls. If you shake the box harder you make the balls move faster and their separation distance increases. This is the microscopic explanation for the macroscopic phenomenon known as thermal expansion. Certainly the effect can be explained in greater depth with a mathematical analysis, but this is the gist of it.

I disagree with the box analogy here. If I have a box full of ping pong balls, the density of the ping pong balls (as well as the mean distance between them) is fixed by the number of balls in the box and the size of the box, no matter how fast they are traveling. What increases in your box analogy is the rate that they impact the wall and the mean momentum change that occurs with each impact. This is the equivalent of heating up a rigid container of gas - the density and mean distance between molecules does not change, but the pressure rises as the molecular velocity increases.

 

[bwana]

I think you cannot just look at a single collision in isolation, and wonder what makes the faster particle go up. The probability of different types of collisions likely depends the gradient of temperature. Faster particles do not just rise individually through slower particles. They can also pass on their momentum and become slower. Also keep in mind that for temperature, the particle movement relative to the bulk movement is relevant, not relative to some frame where the pocket of hot air is moving.

I am trying to understand the concept of buoyancy at the statistical mechanical level. Can you provide insight or equation where position of a particle in a uniform gravitational field is dependent on its kinetic energy?

Or do the same experiment in an accelerating spaceship. What is it about colder particles that pushes them to the floor?

 

[A.T.]

What is it about colder particles that pushes them to the floor?

There are no "colder particles". Their speed changes with every collision. Explaining convection based on a single particle traveling down or up is futile.

 

[Herman Trivilino]

I am trying to understand the concept of buoyancy at the statistical mechanical level.

An object experiences a buoyant force because the particles colliding with the bottom of the object transfer vertical components of momentum to the object at a faster rate than do the molecules colliding with the top of the object.

You can apply this argument to answer your query about a layer of hot air floating above a layer of cold air, but it's complicated by the fact that you don't have a well-defined boundary between the layers like you do between an object and the surrounding fluid.