Freeqs. On Charlottesville’s downtown mall Free Speech Wall, the solution to a differential equation describing motion under uniform gravitational acceleration field (as near the surface of a planetary body; here the Earth with a=~9.8 m/(s^2) ) and air resistance.
Air resistance is approximated as linearly proportional to velocity for simplicity and brevity here, but as far as I know including a quadratic term greatly aids the accuracy of the approximation for higher velocities.
dv/dt = 9.8 - kv
Note that positive displacement and velocity is defined to be down / in the direction of gravitational acceleration in this case. For me this is the easiest / most intuitive way to define the basic relationship.
You can of course make the down direction negative (such that the DE becomes dv/dt = -9.8 + k|v|) but it’s crucial to keep in mind that the acceleration due to drag is directed opposite the direction of motion, and so adding the absolute value of the velocity (which will of course initially be negative) is necessary.
Otherwise the system described will be one where velocity grows without bound exponentially rather than initially growing quickly and then approaching an asymptote as it actually does given the nature of the system.
This is from back in April 2011; I hadn’t gotten around to uploading it from my phone until now.
