given the amplitude of a damped harmonic oscillator drops to 1/e of its inital value after n complete cycles show the ratio of oscillation to the period of the sam oscillator with no damping is

T damped/ T undamped = (1 + 1/(4pi^2 n^2) ^(1/2)

i got the auxillary parts where T undamped equals 4pi squared/q^2 where q is 2pi n , and T damped is 4pi ^2n^2 + c2pi n/m somethings not working because i get the wrong stuff on the bottom and c2pi n/m has to equal 1? or rearranging it i get square root ((cn2 (pi))/k + 1) equals Td/T so cn2 pi /k has to somehow equal 1/(4pi^2n^2)

for the second problem the terminal speed of a freely fallin ball is v when the ball is supported by a light elastic spring the spring stretches an amount x, show the natural frequiecy is

w damped = sqare root ( g/x-g^2 / 4v^2)

i know how to prove w undamped equals squareroot (g/x) that is easy

for this one i am getting confused with the question from the time when it is accelerating to when it hits terminal speed and doesnt accelerate, do i have to integrate at all or set up two parts to the problem one with acceleration and one without? because how do i know if it hits terminal speed before it stretches the spring to the max x?