[Title thanks to the Marathon game series.]
We've (mostly) moved in to our new home in Gainesville, FL! This week's post is actually inspired by something I've noticed about the house's A/C. A few of the vents are mounted near the ceiling, and I can feel where the cold air is hitting the ground a few feet away:
This wouldn't be too surprising for something heavier than air, which would follow a parabolic trajectory, but I assumed the air resistance of air would be pretty high. Since cold air is denser than warmer air, I supposed I could try to find the change in energy as the cold air dropped. Using the gravitational potential energy, we can imagine swapping a bit of cold air for a bit of warm:
where m is the average mass of an air molecule, d is the vertical distance between the two bits, g is the acceleration of gravity, and ρ 1 and 2 are the number of air molecules per unit volume for the cold and warm air respectively. Using the Ideal Gas Law, we can write
where T 1 and 2 are the temperatures (in Kelvin) of the two gases. We can combine these two equations, and use
to get the force on the bit of cold air:
What this says is that the force on the air will be a fraction of the normal gravitational force, determined by the ratio of absolute temperatures. Since F = ma, we can cancel the mass density terms, and find that the acceleration is simply g times 1 minus the temperature ratio.
Let's put some numbers to this to see what kind of ratio we might expect. According to this site, air conditioners typically cool the air in your house by 16-22°F at a time. If we suppose the house is at 75°F when the A/C turns on, we can expect it to put out air at around 55°F. Converting these to Kelvin, we have 297 K and 286 K. That means the cold air will be dropping at a rate of 0.037 * g = 0.36 m/s^2. Assuming our ceilings are about 10 ft high, the air will hit the ground after 4.1 seconds. Without air resistance, it would take only 0.79 seconds for an object to fall, but I had imagined the air mixed long before it ever got to the ground.
We still need to know the speed of the air coming out of the A/C. This site gives the total capacity of a 1-ton A/C unit as 400 ft^3/minute (I have no idea what the tonnage is for ours, but let's go with 1). The vents are about 1 ft x 0.5 ft, and there are 7 in the house, so we get a horizontal speed of about 0.58 m/s. That means in the 4.1 seconds of descending time, the air goes about 2.4 meters from the vent. Without resistance, that would be 46 cm! Often in physics we begin a problem by neglecting friction, air resistance, and/or higher order terms, so it was interesting to take a closer look at a situation where that's not possible.
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