# Same weight but different density equals different speed (terminal velocity)?

Not trying to be more pedantic than LSLGuy, but…

To keep from hijacking this thread, any more than it already has been.

If I understand the theory that’s demonstrated by the "feather and the steel ball in a vacuum’ experiment correctly…
Both will have the same ‘terminal velocity’ (speed for simplicity’s sake) in a vacuum, regardless of the weight of either one.
Correct?

Just for the purpose of fighting my ignorance…
If I had two spherical containers of identical size,
that weighed, oh… lets say 1 gram,
then filled one with enough lead shot to weigh out to a pound.
Then filled the other one with enough feathers to bring that sphere up to a pound.
If both spheres were dropped at the same time, would they both have the same ‘speed’, (terminal velocity) given enough height?
Even if they’re not in a vacuum?

If the spheres are identical size and filled to the same weight, they have the same density.

First of all, there is no terminal speed in a vacuum. Absent of the drag produced by an aerofluid medium, all objects will accelerate at the same rate in the same acceleration field.

For objects falling in an atmosphere, they have a characteristic called the ballistic coefficient (often referred to simply as beta or β). This is the mass of the object divided by the product of the mean aerodynamic area and the drag coefficient. (“Mean area” is used because some objects will tumble and present differing areas, but for most flydown analysis a stable orientation is assumed). Terminal speed for a ballistic object is reached when the drag force at a given altitude equals the acceleration due to gravity; after that, the object will fall “straight down” at a constant rate, and laterally at approximately the same velocity as the atmosphere in that location, usually approximated by a statistical wind model like GRAM-99. For objects with very low β that achieve terminal speed at very high altitudes, you may also need to compensate for the rotation of the Earth, but this isn’t the case for a solid spherical object which should fall fast enough through the upper atmosphere that it will be close to the surface when it reaches terminal speed and will essentially be embedded in the rotating atmosphere along with the Earth’s surface.

The answer to the o.p. is that two spherical containers of the same external dimensions (and therefore the same aerodynamic area and drag coefficient) but containing identical masses (of whatever) would fall at the same rate from any height. The terminals speeds would be the same as they would have the same β.

Stranger

You’re correct. The important value is known as “sectional density”, or the total mass divided by cross-sectional area. (Cross-section in the sense of the part that meets the medium resisting it).
One nitpick - falling in a vacuum, there is no ‘terminal velocity’. The speed reaches a constant value only when there are no net forces (gravity = drag). Your aside is appropriate, as their speeds will match.

Sectional density is not quite the same thing as the ballistic coefficient, and is more useful in comparing the penetration of relatively like shapes through a viscous medium (where the shape of the ballistic object plays only a small contribution in total drag) such as bullets through ballistic gelatin or flesh. This is why an unexpanded 9mm Parabellum or .22 LR will often punch through and through a body while a .45 ACP roundnose will remain in the body. For purely aerodynamic drag, β is a more useful quantity.

Stranger

The OP is talking about identically shaped objects. I was at least partly adding to scr4’s mention of density. Just because SD is used as you describe doesn’t mean it’s not useful in this discussion, given that the drag coefficients are the same.

It should also be pointed out, though I imagine it was assumed by the OP, that the containers are sealed to the air as well.

However, sectional density is used in terminal ballistics as a relative comparison for penetration and travel through a viscous hydraulic or elastic solid medium, whereas the ballastic coefficient is generally used in external ballistics of an object travelling through an aeroelastic medium. The causes and effects of drag in a marginally compressible elastic or incompressible hydraulic medium versus a highly compressible aeroelastic medium are substantially different; hence the distinction. I am sorry if this comes off as being unduely pedantic, but it is very important in considering the relative effects of different objects.

An additional real-world consideration are the mass distribution and resulting perturbances of the various projectiles, especially at terminal speed. Because the lead shot is very dense, it will tend to pool in the bottom of the (presumed rigid) lightweight aeroshell, probably causing some amount of corkscrewing of the shell as it spins about the down-pointing axis, possibly (depeding on the surface roughness) increasing turbulence and permitting a slightly higher rate of descent than the feather-filled aeroshell, which will tend to tumble or roll lateral to the downward path with its more even distribution of low density feathers. Not all masses are the same, even if they fit within the same external shape.

Stranger

First things, first. I thank all of y’all for taking the time to try and “edumacate” an old ‘wood butcher’.

Even if the substance that is used to fill one of them is not the same as what is in the second one?
I don’t mean to be obtuse, but I thought density had something to do with the solidity of a substance. Would this simple example be correct or not? A solid, square piece of wood, is the same density as a solid, square piece of iron. Both objects having the exact same dimensions.

Does that mean that, in the absence of an atmosphere (in a vacuum) the speed (velocity?) of an object would steadily increase, until it impacted whatever it is drawn to by gravity, with the objects speed wholly dependent upon the distance travelled?
(the farther it ‘falls’, the faster it’ll be going when it impacts?)

You are correct sir, I assumed them to be a hollow sphere, each one exactly the same shape, size and weight (empty hollow sphere = 1 gram), sealed and perfectly smooth, with the only difference being the contents.
One would contain 1 pound (minus 1 gram) of lead shot, the other would contain 1 pound (minus 1 gram) of feathers.

So both of these statements are correct? (Albeit, tremendously over-simplified.)
Both of the spheres would have identical speeds, the only determining factor would be drag induced by an atmosphere. The balls will reach a maximum (terminal) speed, in an atmosphere.
The balls will have identical speeds, that steadily increases until impact, in a vacuum.

<Ow! My brain hurts, now. Going to get an aspirin…>

Density is just mass per volume. So in your wood vs. iron example, the two objects do *not *have the same density: they both have the same volume (because they have the same dimensions), but iron is heavier than wood (more massive), so the iron cube is more dense than the wood cube.

In your original sphere example, it might help to think of the “*average *density” of each sphere (meaning you look at the *total *mass and volume of sphere and its contents) as distinct from the density of the sphere’s contents.

Consider what you’d have to do to actually make two spheres filled with feathers and lead shot that weighed the same. The sphere filled with feathers would probably be stuffed full of feathers, while the one containing the shot would have just a little bit of lead shot rattling around inside the hollow sphere. That’s because the density of feathers is much less than that of lead, so it takes more volume of featherss to get the same mass. However, by your rules, the *total *mass of each sphere is the same (1 pound), and the total volume is also the same. Thus the “average density” of each sphere, accounting for the sphere itself, its contents, and any empty space inside the sphere, is exactly the same.

Yes.

To expand, there’s a gravitational force on the object. When dropped in a vacuum, the gravitational force is the *only *force on the object, so it just keeps going faster and faster. In an atmosphere, air resistance *also *provides a force, which increases with velocity. When the air resistance force equals the gravitational force (and thus the total force is zero), the object’s speed will not change. That’s terminal velocity.

Density is more simple than this. It is Mass / Volume. That’s it. Take an object, measure its mass, measure its volume, calculate the density. You have two identical spheres, inside of each you have 1 pound of material. Same total mass, same total volume, same density.

Your wood example is incorrect. A solid piece of iron has more mass (is heavier) than a solid piece of wood of the same size, so it has greater density

In a way… the movement of objects is governed by Newtons second law. Force = Mass x Acceleration. In your vacuum example, the ONLY force is the force of gravity. So your object will continue to accelerate until it hits something, and gets acted on by another force.

Terminal velocity represents a balancing of two forces, gravity and air resistance. Once the object is going fast enough that the force from air resistance is exactly equal to the force of gravity, they cancel each other out and the total force on the object is 0, so the acceleration is 0, and it stops going faster, and will keep going that speed until another force acts on the object, like from the ground.

This is certainly correct. You could nitpick it by noting that the force due to gravity increases very slightly as the object falls; drag also increases (and not so slightly) due to the density of air increasing. The net result is that the falling object does not keep going the same speed - it actually slows down gradually.

Another nitpick:
It would be quite challenging to come up with a 1-gram spherical container large enough to contain 452.6 grams of feathers that was sufficiently rigid not to be distorted when falling through air. The distortion would not be identical to that of a similar sphere containing lead shot, so the two should not be expected to exhibit the same terminal velocity.

Trivia:
Q: Which weighs more - a pound of feathers or a pound of gold?
A: A pound of feathers. It is measured in British units, 16 oz., while precious metals are measured inavoirdupois where a pound is 12 avoirdupois ounces.

Lead, as a base metal, I assume the OP is talking apples and apples and measuring in British units.

Density is measured in comparison to water ( =1). Iron IIRC is about 6, or six times heavier. Feathers are very fractal, so a lot of volume is air; unless you compress it with a hydraulic press until you have pure feather material, not much different than hair or fingernails, about the same as water give or take.

(When you shed hair in the bathtub, it does not sink to the bottom once it breaks surface tension; it sort of meanders around on the currents because it is about the same density, a bit heavier. For yuck factor, I recall a public pool that had not been propery cleaned - there were giant hairball dustbunnies rolling around the deep end, very thin but about 3 feet in diameter.)

But the posters above are correct. Terminal velocity is air drag vs. pull of gravity. It is reached when the forces balance. Drag is determined (very roughly) by weight vs, cross-section; with a factor for “streamlining” as shape can reduce drag. In a vaccuum, there is nothing slowing you down; which is why the Apollo capsules coming in from the moon hit some VERY high velocities before reaching the atmosphere, where drag converted velocity to heat (and air turbulence) and slowed them down.

Try this simple experiment- drop a flat piece of paper; crumple it up and drop it. Fold it into a very dense, very small lump and drop it. If necessary, videotape dropping these out of a second floor window and count the frames to get approximate drop times. Same mass, only cross sectional air resistance differs. Same density because it’s the same material…

Going back to my original pedantic statement about a bag of lead & a bag of feathers. Implicit in that was the thought

Now drop them. The lead one is the size of a golf ball. The feather one is the size of a beach ball. They experience the exact same force due to gravity but very different forces due to aerodynamic drag. The lead bag wins the race by a good margin assuming they fall farther than a few inches.

Not sure if I should take that as a compliment, a swipe, or just a funny turn of phrase …

That’s all kinds of jacked up. First of all, an avoirdupois pound is a British pound, and that’s the one that’s 16 oz. The other one is the Troy pound, which is used for precious metals and is 12 oz. But it doesn’t stop there, because an avoidupois ounce isn’t the same thing as a Troy ounce. The former is 28.35 metric grams while Troy is 31.1 grams, making the Troy ounce bigger and an ounce of gold heavier than an ounce of feathers.

But there’s only 12 of these heavier units, and in the end, you get 373g for Troy and 453.6g for avoirdupois pounds.

So remember that- if the question says ounces, it’s gold. If it says pounds, it’s feathers.

Doh! Too early in the morning. Yes, Troy ounces…

At the risk of being more pedantic ;), I’d like to point out a counterexample to the solidity idea. Ice is more solid than water, but has a lower density. That’s why it floats.

Once again… Ignorance, successfully fought!

Thank you all, your efforts have not been in vain.

Most definitely not a ‘swipe’, sir.

I rather like, ‘a complimentary, funny turn of phrase’.

Similarly, many people would tell you that a pint of cream is heavier than a pint of milk… but cream floats on milk, doesn’t it?