Sign error for buoyant displacement of a Champaign bubble and possible solution

[Wiljea]

In the example 2 of "Buoyancy & Terminal Velocity" ("Motion" section of the equation library), we find the following solution:
[ vt=-1.14234 81034 5 m/s v=-0.79446 37698 68 m/s tfr=0.21337 88142 9 s xfr=0.15488 56277 51 m ]
where the first 3 values are the right ones (but not the fourth) corresponding respectively to:

• vt < 0 which is the expected negative value of the terminal velocity for an ascending bubble
• v < 0 which is the expected negative speed achieved after t=0.1 s
• tfr the time required to reach the fraction fr=0.95 of vt
• xfr the displacement during tfr which should be negative here corresponding to an ascending motion.

The problem comes from a probable error in the fourth equation which is an integral of the hyperbolic tangent:
"'xfr_ft=(vt_(m/s))∫(0_s;(tfr_s)/(1_s);TANH(tⒸg/vt_(m/s));t)' "

where vt (which is negative) appears at 2 places. And since TANH is an odd function, it appears clearly that the result will always be positive. This means that one of the 2 vt should be in an absolute value function. And it should be the one appearing in the TANH argument because this is a time integral which should give a positive duration:
Δt = ∫f(t) dt > 0. This should also be corrected in the reference https://en.wikipedia.org/wiki/Terminal_velocity.

In our case this integral needs to be corrected accordingly in 2 places, since it appears also in the previous sub-section "Terminal Velocity" as well as in "Buoyancy & Terminal Velocity", the equation becoming:
"'xfr_ft=(vt_(m/s))∫(0_s;(tfr_s)/(1_s);TANH(tⒸg/ABS(vt_(m/s)));t)' "

Strangely, it did this correction in an old version of equations.cc. But, now I just checked the last version 0.9.2 (branch "stable") and both corrections are missing. Therefore it is now done in the following:
Update equations_JW_25_03_09PM1.cc
stable...Wiljea:db48x:patch-25.