newtonian mechanics - Free Fall with Air Resistance

Can somebody help me understanding why when free falling the force on the vertical axis is F= -mg-kv, where k is the constant of air resistance and v the velocity?

Suppose the vertical axis is positive in the upwards direction. Then the acceleration is negative, so this is okay. The air resistance should be positive thou, cause it should oppose to the acceleration. Am I right or wrong? Cause I don't understand how the air resistance and the acceleration could ever have the same sign. Indeed when opening a parachute, all we do is enhancing the air resistance and therefore balancing the two forces.

Answer

A general model of restive forces that depend on velocity can be given by the following equation,

$F = C_1v + C_2v^2$

We choose either the force to be dependent on either of the terms or both depending on the magnitude of velocity.

In your case, since we are talking about bodies falling we can assume the velocity to be small and ignore the $v^2$ term.

The resistance force always oppose the motion of a body.

The body you are talking about is falling down due to acceleration due to gravity. Since restive forces always tend to bring the body in motion to rest, the restive forces will act upwards in an attempt to slow down the object.

So we have two forces, gravity which pulls the object down and the air resistance which tries to slow down the object.

Writing the equation of motion for the body using your sign convention (upwards as positive), we get

$F = C_1v - mg$

The vector form of the equation would be

$F = -C_1\vec{v} - m\vec{g}$

Note that the velocity and acceleration are in opposite directions hence the additional negative sign.

The signs of the two forces are in opposite direction which is in agreement with out day to day experiences and intuition.

An object is said to be in free-fall if its motion is caused by gravity only. Your case where air provides resistance, the object in motion cannot be said to be in a free-fall.

(Video-Lecture) 12: Resistive Forces | 8.01 Classical Mechanics, Fall 1999 (Walter Lewin)