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I would like to know why when a body falls with air resistance, we have that the drag caused by the air is $-kv$ and the acceleration is $-mg$.

What I don't understand is how they both have the same sign $F_{\text(gravity)}=-mg$ and $F_{\text(drag)}=-kv$.

I understand that the drag is negative cause is opposed to the velocity. I understand that the $-mg$ part is negative cause the vertical axis is positive in the upward direction. But I don't understand how come that air resistance has the same direction of the gravitational force. If I fall, I should feel an acceleration going down, and hence there's a force poiting downwards, and I should also feel a force going upward, cause of the air that I go through.

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You are right that for a falling object, the force of gravity is down while the drag force points up (in the direction opposite to the velocity). And I understand your confusion from the equations - why is the force of gravity $-mg$?

This is because we usually write $g$ as a positive number, without taking account of its direction (even though it is pointing down). But when we write the drag force as $-kv$, we imply that $v$ is a vector $\vec{v}$ which therefore has a direction as well as a magnitude - and the drag force, which is also a vector, must point in the opposite direction.

It would be more consistent if we wrote acceleration as a vector $\vec{g}$, in which case the minus sign would disappear. But that is not the convention...

This means that your best approach is a diagram that shows, for a particular situation, the direction of the forces. This will allow you to get the right answer even in harder problems (for example, the case where a projectile is initially launched upwards, in which case both gravity and drag point down...).

Using an equation "blindly" (that is, without considering the sign conventions) can be very dangerous. By asking this question, you have shown that you are aware of this danger. That's a very important step to mastering this class of problems.

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