Question

Determine a differential equation for the velocity v(t) of a falling body of mass m if air resistance is proportional to the square of the instantaneous velocity. Assume the downward direction is positive. (Use k > 0 for the constant of proportionality, g > 0 for acceleration due to gravity, and v for v(t).)

Answer 1

A freely falling mass m will be pulled down toward the ground with downward acceleration +g while feeling upward drag D due to air resistance with acceleration -a such that D = - m a = - k v ². Hence the body's velocity v(t) changes with respect to time t according to the differential equation,

dv(t)/dt = m g - k v ²