Explanation:
When a body or object falls, basically two forces act on it:
1. The force of air friction, also called "drag force" [tex]D[/tex]:
[tex]D={C}_{d}\frac{\rho V^{2} }{2}A[/tex] (1)
Where:
[tex]C_ {d}[/tex] is the drag coefficient
[tex]\rho[/tex] is the density of the fluid (air for example)
[tex]V[/tex] is the velocity
[tex]A[/tex] is the transversal area of the object
So, this force is proportional to the transversal area of the falling element and to the square of the velocity.
2. Its weight due to the gravity force [tex]W[/tex]:
[tex]W=m.g[/tex]
(2)
Where:
[tex]m[/tex] is the mass of the object
[tex]g[/tex] is the acceleration due gravity
So, at the moment when the drag force equals the gravity force, the object will have its terminal velocity:
[tex]D=W[/tex] (3)
[tex]{C}_{d}\frac{\rho V^{2} }{2}A=m.g[/tex] (4)
[tex]V=\sqrt{\frac{2m.g}{\rho A{C}_{d}}}[/tex] (5) This is the terminal velocity
As we can see, there is no "falling time" in this equation.
Therefore, the terminal velocity is not dependent on the falling time.
