(24)
this shows that the velocity u will send asymptotically to a certain limit V (called the terminal velocity) such that kV2 = g. The solution is
| u = V tanh | gt | , x = | V2 | log cosh | gt | , |
| V | g | V |
(25)
if the particle start from rest in the position x = 0 at the instant t = 0. In the case of a particle projected vertically upwards we have
| du | = −g − ku2, |
| dt |
(26)
the positive direction being now upwards. This leads to
| tan−1 | u | = tan−1 | u0 | − | gt | , x = | V2 | log | V2 + u02 | , |
| V | V | V | 2g | V2 + u2 |
(27)