As I understand it, terminal velocity is when atmospheric friction balances the acceleration due to gravity on an object in freefall over something with an atmosphere. It’s modified, obviously, by the shape of an object: Something like a lawn dart pointing straight down would have a much higher terminal velocity than a sheet of metal with the same mass.
Thus, it seems possible that some objects will never stop accelerating in the Earth’s atmosphere regardless of how high they are dropped from. (Imagine a space elevator stopped at the very edge of the atmosphere, wherever you take that to be.) They are so aerodynamic the force of friction cannot become great enough to stop the acceleration due to the force of gravity acting upon the object.
For object much smaller than the Earth, it’s sort of theoretically possible, but not going to happen. Remember, air resistance goes up with the square of the velocity, so the force goes up quickly as an object speeds up. And if my calcs are correct, something falling from one definition of the edge of the atmosphere (100 km), is going to be going about 3,000 mph when it reaches the ground (if there was no air resistance). It’s hard to imagine any reasonable-sized object that isn’t going to get more wind resistance force than gravity force when it’s going 3,000 mph through sea-level atmosphere.
[OK, maybe some science-fictional object made of exotic super-dense matter might do it, but nothing believable in the forseeable future).
Now, if a giant moon-sized stony asteroid smashed into Earth, the atmosphere might not slow it down all that much, but I’m not sure ‘terminal velocity’ has a whole lot of meaning in that case.
Theoritically, an object in freefall never actually reaches its terminal velocity (defined as the speed at which drag force equals weight); the acceleration slowly approaches zero as the object picks up speed.
It might be more useful to set a hard limit, and carefully define the problem, like this:
is it possible to have an object with mass and drag characteristics such that when dropped from the edge of space (100 km above sea level), it will achieve 95% of its terminal velocity before reaching sea level?
The answer is yes. Roughly speaking, drag scales with the cross-sectional area of an object, while weight scales with the volume. Area scales as the square of radius, while volume scales with the cube of radius. In a nutshell: if your object has too low a terminal velocity, make it bigger.
Just off-the cuff estimates here: A steel BB (0.177 inches diameter) will likely hit 95% of its terminal velocity after a fall of 100 feet or so. A solid steel sphere the size of a bowling ball (~8 inches diameter, ~70 pounds) will probably fall a couple thousand feet before hitting 95% of its (much higher) terminal velocity.
Still not happy? Make the steel sphere larger. Try a ten-foot, solid-steel sphere, at around 230,000 pounds. I haven’t done the math, but if you could somehow hoist that pig up to the edge of space and drop it, I suspect it might possibly hit sea level before achieving 95% of its terminal velocity. If not…well…make the sphere bigger. Streamline it. Make it a solid steel cylinder with fins and a nosecone. My gut feeling is that it would in fact be possible to have a reasonably sized/shaped object that could fall 100 km through the earth’s atmosphere without coming close to its terminal velocity.
I have an excel spreadsheet model of the standard atmosphere, with pressure/density/temperature up to the edge of space. Maybe one of these days I’ll run a sim of a falling object and try different mass/drag characteristics to see what happens…
How do you figure? Unless you’re thinking some kind of Zeno’s Paradox (in order to hit the terminal velocity of X, first you have to be going 0.5X, then 0.75X…etc) there are plenty of objects that have a terminal velocity easily reached under “normal” conditions.
Scroll down until you see “Derivation of the solution for the velocity v as a function of time t” in small print, and click on “show”. Look for the equation near the bottom of that box, showing velocity as a function of time.
Note the text immediately below that equation:
IOW, a body in freefall through the atmosphere asymptotically approaches terminal velocity. This must be so, because as you approach terminal velocity, the net accelerative force approaches zero. If you’re still unconvinced, consider plotting that equation in Excel, and try to idenfity the time at which v matches the terminal velocity calculated by the expression right below that quote.
I’ll grant that it may be possible for small/light objects (such as a BB) dropped from the edge of space to exceed their sea-level terminal velocity before reaching the dense lower levels of the atmosphere (since aero drag up high is pretty small), and then at some point begin decelerating; the crossover point would be the particular instant where its speed exactly matched the terminal velocity at that particular altitude. But this exception is only because of the varying density of the atmosphere with respect to altitude.
If you run the equations, you’ll find that, indeed, things only approach terminal velocity and never actually reach it. It’s not Zeno’s Paradox, because that just has an infinite number of “steps”, whatever those are, but all packed into a finite amount of time. But an object in free-fall really does take an infinite amount of time, not just an infinite number of “steps”, to reach terminal velocity.
Of course, the real world doesn’t exactly follow the nice textbook equations. In the real world, there are things like air currents, Brownian motion, and quantum indeterminacy, which will cause things to actually reach their terminal velocities. But those are really hard to deal with in the equations, so we usually just ignore them.
Better still, try iridium or osmium (so close in density that we won’t quibble the difference). About twice as dense as lead, or nearly three times as dense as steel. Streamline it and make it big enough that the square/cube law works in it’s favor and it just might work. Also both those metals have high melting points which would be desirable to cope with frictional heating.
Claim: An object never quite reaches terminal velocity. To prove it, here’s the equation which describes the object’s motion.
Counterargument: Here are some real-world conditions which the equation doesn’t take into account which mean that terminal velocity is actually reached.
Defender of the claim: But they are too hard to factor into the equation, so we don’t bother. Therefore the proof stands.
Not really. If you didn’t make any assumptions you wouldn’t be able to determine anything - you would need excessively complex equations with more input information than you can know for certain. What you can do is figure out what order the effects of Brownian motion, air currents, etc will have on the falling object. I’m pretty sure that they will not be on the order of the force that air at 3000 mph (I’m assuming Quercus is correct on that calculation) will impart on the object so they can then rightfully be neglected while retaining reasonable accuracy.
The basic form of the velocity-versus-time curve is an arc starting at zero and asymptotically approaching TV as time goes to infinity. All of those real-world effects are basically static (random noise) superimposed on that curve. If you’re trying to answer the question “when does object X dropped from height Y reach TV?”, the noise means you’ll get a different answer every time, cuz it’s…noise. In fact, if we’re going to shitpick on that small of a scale, then I propose that the very definition of “terminal velocity” is meaninless: it fluctuates with every little breeze, every fly-fart, every tiny little air current and molecular impact and local density fluctuation that our projectile experiences during its fall.
It seems to me that the criterion should not involve a frictionless drop from “the edge of the atmosphere”, but rather the limit as the height from which it is dropped goes to infinity (but ignoring bodies other than the Earth, such as the sun). Correct me if I"m wrong, but I believe this limit exists and is equal to the Earth’s escape velocity at its surface, which is 11.2 km/s (about 25,000 mph by my calculation).
I suppose we could ask about things whose terminal velocity is greater than c, but I have no idea whether that really makes sense.
You’re right about infinite-height drop resulting in a max possible sea-level velocity of ~25K mph. In fact, an object falling from extreme heights will have achieved most of this velocity before it encounters the atmosphere; if it’s a really big, really dense object, such that its calculated TV is greater than 25K MPH, it may continue accelerating as it falls through the atmosphere, all the way to sea level.
This is not to say that the object does not have a terminal velocity - indeed, for a simple shape, it should be calculable for any sized object - only that the object will never achieve its TV in freefall, no matter from how high.
