A point performs dampled oscillations according to the law `x=a_(0)e^(-betat)sin omegat `. Find `:`
`(a)` the oscillation amplitude and the velocity of the point at the moment `t=0`,
`(b)` the moments of time at which the point reaches the extreme positions.

Explanation

We are given `x=a_(0)e^(-betat ) sin omega t `
(a) The velocity of the point at `t=0` is obtained from
`v_(0)=(dot(x))_(t=0)=omega a_(0)`
The term ' oscillation amplitude at the moment `t=0"` is meaningless. Probably the implication is the amplitude for `t lt lt (1)/(beta)` . Then `x~~ a_(0)sin omega t ` and amplitude is `a_(0)`.
(b) `dot(x) =(-betaa_(0)sin omegat + omega a _(0) cos omega t ) e^(-betat)=0`
when the displacement is an extremum. Then
` tan omega t =(omega )/(beta)`
or `omegat=tan ^(-1)((omega)/(beta))+n pi, n=0,1,2,........`