where u0 is the velocity of projection. The particle comes to rest when
| t = | V | tan−1 | u0 | , x = | V2 | log ( 1 + | u02 | ). |
| g | V | 2g | V2 |
(28)
For small velocities the resistance of the air is more nearly proportional to the first power of the velocity. The effect of forces of this type on small vibratory motions may be investigated as follows. The equation (5) when modified by the introduction of a frictional term becomes
ẍ = −μx − kẋ.
(29)
If k2 < 4μ the solution is
x = a e−t/τ cos (σt + ε),
(30)
where