An object with mass is dropped from rest and we assume that the air resistance is proportional to the speed of the object. If s(t) is the distance dropped after t seconds, then the speed is v = s(t) and the acceleration is a = v '(t). If g is the acceleration due to gravity, then the downward force on the object is mg - cv, where c is a positive constant, and Newton's Second Law gives
m dv / dt = mg - c v
(a) Solve this as a linear equation to show that
v = mg / c (1 - e-ct / m)
(b) What is the limiting velocity?
(c) Find the distance the object has fallen after t seconds.