# Hyperbolic tangent function for terminal velocity with Vo>Vt

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1. Mar 3, 2015

### birk

Hi! First post on this forum, though not the first time visiting :)

I am working on a model of an object falling from one layer of air density into another layer with a higher density, effectively changing the acceleration from positive to negative instantly. (Somehow I am thinking of positive as the downwards direction here). The transition between the two "zones" is instantaneous. (One way to think of it could be an object (that sinks) that falls at a high velocity into water and is abruptly decelerated.)

My question is: How does one calculate the speed of the object after it has entered the new medium? I have seen the following equation for terminal velocity on wikipedia and other sources:

but this assumes an initial velocity of v_0=0

In my situation the initial velocity when entering the new medium is nearly the double of the terminal velocity that I have calculated from the first part:

One solution I tried, which I quickly realised how stupid was, was to multiply the function by -1, but this left me with a terminal velocity that increased when the drag coefficient was increased, which is not really what one wants.

Any input would be very much appreciated!

2. Mar 3, 2015

### mathman

I have no background for this subject. However, it seems strange to me that the terminal velocity depends on t (which I presume is time).

3. Mar 3, 2015

### birk

To answer your question: You are right, the terminal velocity does not depend on t (time). It depends in the various aerodynamic properties of the object falling and the medium that object is falling through + initial velocity. The definition of terminal velocity is that the forces from aerodynamic resistance (drag) are equal to the gravitational forces pulling the object down, effectively rendering a=0 and a constant, "terminal" velocity.

My question is rather regarding the velocity of an object tending towards a terminal velocity from a set t=0 with an initial v0.
For the above equation to work properly, it seems as if V0 HAS TO BE 0, which is obviously not working in my scenario. I found something that could indicate a possible answer:

https://books.google.no/books?id=iYALAAAAQBAJ&pg=PA188 (First third of the page, but it is somewhat unclear to me...)

but I feel I am in deep waters on this, so any help would be hugely appreciated!

4. Mar 4, 2015

### mathman

Based on elementary considerations I don't understand why the initial velocity would matter. In free fall objects usually start at zero velocity and speed up due to the acceleration of gravity until terminal velocity is reached.