Chapter 12 - Kinematics of a Particle - Section 12.2 - Rectilinear Kinematics: Continuous Motion - Problems - Page 19: 27

$t=0.549(\frac{v_f}{g})$

$\frac{dv}{dt}=a=(\frac{g}{v_f^2})(v_f^2-v^2)$ $\int_0^v\frac{dv}{v_f^2-v^2}=\frac{g}{v_f^2}\int_0^t dt$ $\frac{1}{2v_f}\ln(\frac{v_f+v}{v_f-v})|^v_0=\frac{g}{v_f^2}t$ $t=\frac{v_f}{2g}\ln(\frac{v_f+v}{v_f-v})$ $t=\frac{v_f}{2g}\ln(\frac{v_f+v_f/2}{v_f-v_f/2})$ $t=0.549(\frac{v_f}{g})$