Mechanics question derivations terminal velocity (quadratic case)

[Genericcoder]

hi so basically my question is derivation of derivation of the quadratic case of the terminal velocity


I have attached the pictures in picture 1 I don't know I think the book has error in his derivation @ equation 2.4.13 shouldn't it be + instead of negative ??

& in second picture their is something I don't understand two stuff acctually first is in equation 2.4.15
how did he come up with 1/2 * dv^2 / dy? and my second question is
du/dy = -1/vt^2 * dv^2/dy how did he come up with dv^2/dy ??

thanks alot.

 

[maajdl]

Yes the sign is wrong.
And dv/dt = dv/dy dy/dt = dv/dy v
since v = dy/dt .
And since dv²/dt = 2 v dv/dt, the last result follows.

 

[Genericcoder]

could u explain this line a little more ->dv²/dt = 2 v dv/dt, the last result follows?

 

[maajdl]

read:

dv²/dy = 2 v dv/dy

which implies

v dv/dy = 1/2 dv²/dy