Given an object of mass m moving in a resistive medium, suppose that the drag per unit mass is proportional to speed (with constant of proportionality ), and that there is no other force acting on a body. Then the equation of motion is given by
Solving the differential equation using the initial conditions and gives
(shown above in red). Differentiating then gives the speed as
(shown above in blue). The speed of the body thus decreases to zero over time, and the body reaches a maximum terminal distance
denoted with a dashed line in the plot above.
Free Fall, Terminal Distance, Terminal Velocity
References
Jeffreys, H. and Jeffreys, B. S. "Motion of a Particle Under Gravity with Resistance Varying as the Velocity." §2.11 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 77-78, 1988.
