# Have multiple terminal velocities for a rigid body ever been verified experimentally?

The coefficient of drag can drop suddenly for some objects at high Reynolds numbers. If the drop is "fast enough" then the total drag force, $$F=1/2 \rho A v^2 c_d$$, could actually drop. This occurs whenever $$d(c_d v^2)/dv = 2 c_d v + v^2 d(c_d)/dv < 0$$. Simplifying the equation a bit and using the Reynolds number in place of $$v$$ (since all the other factors are constant), we have: $$d(c_d)/dN_{re} <-2 c_d/N_{re}$$. If this true for any $$N_{re}$$, an object with the correct weight will have (at least) two (stable) terminal velocities, assuming that the orientation remains fixed.

Falling objects will reach their lowest terminal velocity. However, if they are launched downward at high speed they will slow down to their highest terminal velocity.

A completely different way for an object to have multiple terminal velocities is to be stable falling in two different orientations. This is because $$c_d A$$ is in general orientation dependent.

Have multiple terminal velocities for a rigid body ever been verified experimentally? It may be easier to use water since higher Reynolds numbers are reached at lower velocities.

• Even a rigid body can have different cross-sections at different orientations. Are you trying to ask about an object that has an orientation independent cross section?
– hft
Dec 20, 2022 at 21:25
• Like are you asking whether it is possible for a spherical object to have two different terminal velocities?
– hft
Dec 20, 2022 at 21:25
• @hft: There are two non-trivial questions I am asking. 1. Different cross-sections with a stable equilibrium for two or more different orientations. 2. Different Reynolds numbers without changing orientation. There is no obvious way to build either shape, so both questions are interesting. Dec 20, 2022 at 22:00

Before deploying a parachute, a person typically lies horizontally face down. Terminal velocity is about $$120$$ mph.

To go faster, a person reorients himself head down. Terminal velocity is about $$180$$ mph.

Given your comment, it appears you are interested in rigid object that can fall stably in different orientations.

It isn't true that an object necessarily will fall in any particular orientation. Snowflakes can stably tumble as they fall, and there can be more than one stable mode. It depends on shape among other things, and there can be more than one stable mode for a given shape. I don't recall details.

Or perhaps a wing is what you are asking about? If flies stably at high speed, but stalls if too slow?

• Humans are not a rigid body. Different conformations make different orientations stable. Dec 18, 2022 at 3:34
• In OP's defense, even in the very first version of the question (prior to editing), OP does explicitly state: "Has multible [sic] terminal velocities for a rigid body ever been verified experimentally?" (Emphasis added). So, it's not really just in the comments that OP says they are interested in a rigid body...
– hft
Dec 20, 2022 at 20:56
• Therefore this post does not really answer OP's question... probably best to remove it...
– hft
Dec 20, 2022 at 20:57
• On the other hand... pretty much any rigid body other than a sphere could have multiple terminal velocities, depending on the orientation with which the body was dropped...
– hft
Dec 20, 2022 at 20:59
• @hft - True. Imissed the boat at first. But how about the wing? Or perhaps aircraft to provide stability. It is stable in flight and in various stalled modes. Dec 20, 2022 at 21:02