I was talking with a friend about skydiving today (not for me thanks!), when the subject of terminal velocity came up. I then got to thinking about water and that there must be some type of limit there as well. Is there a formula I can use to figure that speed for a given weight, water density, etc…?
What’s the terminan velocity of a hot air baloon in air?
Boyancy becomes much too big a factor to be ignored.
Terminal velocity depends on the body and the fluid. A human body, a house fly and an M1 tank all have different terminal velocities in air… . . and in water. In water a human body will have a very small or zero terminal velocity. If it floats the speed is zero.
Well, that makes sense. Is there a way to figure the speed for, say a 100lb. block of lead?
What shape is the block? The fluid drag forces in water is much more higher than air (it’s ~1000X denser), so your shape is going to be critical.
What you want to do is look up the drag force correlations for your shape, then relate them to the force of gravity (adjusting for bouyancy). Adjust the velocity value until the drag force equals the weight.
I can’t find any correlations for external turbulent flow (all my books are for internal flow), and the only helpful correlation I’ve got on hand is for laminar flow over a sphere. Perhaps some Mech Eng Dopers can lend a hand?
If they ever perfect THIS technology, it would be pretty damn fast…
Wow… This is much more complicated than I had anticipated.
Thanks for the answers, but I think this is more than I really want to get into!
yeah, that’s due to drag… in a vacum, they all have the same, however…
(Mech Eng. Doper)
Unfortunately, it’s not a single, simple formula. As Rabid_Squirrel alluded to, drag force (and terminal velocity, therefore) are proportional to fluid density, object shape, and speed. In air, weight is not usually a factor because the buoyant force is negligibly small, but you can’t make the same assumption for water. Also, since the drag coefficient (a proportionality constant) is usually unknown, it has to be an iterative calculation.
It’s not all bad news, though. My text has a comparative study on terminal velocity of a 50-mm plastic sphere falling in air and water. They determined the velocity in air to be 41 m/s (over 90 mi/hr!), while in water the same sphere would fall at 0.63 m/s (1.4 mi/hr). I just don’t feel like doing the math now for your 100-lb block of lead. Maybe later…
In a vacuum, there is no terminal velocity. Terminal velocity is defined as the point at which the acceleration due to drag exactly counters the acceleration due to gravity. In a vacuum where this is no drag or other forces on the body except gravity, the body continues to accelerate without limit.
Please don’t! That was just something I threw out there to get a rough idea of the difference. Like I said, it was just something that crossed my mind and I thought there might be some simple answer. Alas, I think I’d be better off asking about the meaning of life!
Thanks again,
whatami
As far as I remember, water produces 600 times more drag than air; you can begin there.
Boyancy is a big issue too, specially for a human; in shallow water a body does usually have positive boyancy, but as you go down the water pressure compress your cavities (not only the ones in your teeth) so the body actually becomes more dense. Free divers deal with that all the time.
Sorry that you think it’s so hard. It’s actually a pretty simple concept to understand, but it’s difficult to get exact numbers. Hopefully somebody can correct me if I make any huge errors, but here’s how you get an approximate value.
The formula for downward force is:
F = (m - rho × Volume) × g
(Note: rho is the density of the medium, in this case water.)
The simple formula for drag force is:
D = 1/2 × C[sub]D[/sub] × rho × V[sup]2[/sup] × Area
This formula is not so good if you wanted to be precise, but from the sound of your OP you just want a general idea. Everything in there is a constant except for the speed V, and that’s what you want to solve for. For terminal speed, you set D = F, and get:
V = sqrt(2(m - rho × Volume) × g / (C[sub]D[/sub] × rho × Area))
This simplifies to:
V = sqrt(2(rho[sub]OBJ[/sub]/rho - 1) × g × L / C[sub]D[/sub])
L is the characteristic size of your object (Volume / Area), and rho[sub]OBJ[/sub] is its density.
Now, the only actually complicated part is C[sub]D[/sub]. Getting an exact value for this is tough, as it depends on the shape and everything. But the values don’t vary all that much if you’re dealing with unstreamlined slabs - it’s usually between 0.5 and 1.
So say you’ve got a 0.13-meter slab of lead, which is 11 times denser than water. For C[sub]D[/sub] = 0.5 (spherical-ish), V is around 7 m/s.