Classical Mechanics: One Dimensional Motion with Linear Drag

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What is the motion of an object with a drag force that is proportional to the velocity? Here is my explanation.

This is part of my classical mechanics course. You can find all the videos in this playlist.

00:00 - Intro
02:03 - Newton's second law
03:25 - Differential Equation for Velocity
07:37 - Differential Equation for Position
12:47 - Plotting position and velocity
20:04 - Numerical Calculation of Motion
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I'm attempting to repeat this problem with an added complication: Ambient wind. assuming the wind is constant (2 m/s) and acts in the x direction, how would this effect the derivation of the one-dimensional equations? The ambient wind would apply a constant force, rather than a changing force (as drag force changes with velocity). Would it be removed from the integral as a constant? Much appreciated and great video explanation.

jim