A point performs damped oscilaltions with frequency `omega` and dampled coefficient `beta`. Find the velocity amplitude of the point as a function of time `t` if at the moment `t=0`
`(a)` its displacement amplitude is equal to `a_(0),`
`(b)` the displacement of the point `x(0)=0` and its velocity projection `v_(x(0)=dot(x_(0))`

Explanation

We write `x=a_(0)e^(-betat)cos (omegat+alpha)`
`=Re Ae^(-betat+ iomegat), A=a_(0)e^(i alpha)`
`dot (x) =Re A (- beta+iomega) e ^(-betat+iomegat)`
Velocity amplitude as a function of time is defined in the following manner. Put `t=t_(0)+tau`, then
`x=ReAe^(-beta(t_(0)+tau))e^(iomega(t_(0)+tau))`a
`~~ReAe^(-betat_(0))e^(iomegat_(0)+tauiomegatau)~~ReAe^(-betat_(0))e^(iomegat)`
for `tau lt lt (1)/(beta)` . This means that the displacement amplitude around the time `t_(0)` is `a_(0)e^(-beta t_(0))` and we can say that the displacement amplitude at time `t` is `a_(0)e^(-betat)`. Similarly for the velocity amplitude.
Clearly
`(a)` Velocity amplitude at time `t=a_(0)sqrt(beta^(2)+omega^(2))e^(-betat)`
Since `A(-beta+iomega)=a_(0)e^(ialpha)(-beta+iomega)`
`=a_(0)sqrt(beta^(2)+omega^(2))e^(igamma)`
where `gamma` is another constant.
(b) `x(0)=0+ReA=0 ` or `A=+- ia_(0)`
where `a_(0)` is real and positive.
Also `v_(x)(0)=dot(x_(0))=Re+-ia_(0)(-beta+iomega)`
`=+- omega a_(0)`
Thus `a_(0)=(|dot(x_(0))|)/(omega)` and we take `-(+)` sign if `x_(0)` is negative `(` positive `)`. Finally the velocity amplituce is obtained as
`(|dot(x_(0))|)/(omega)sqrt(beta^(2)+omega^(2))e^(-betat)`