At the moment `t=0` a point starts oscillating along the `x` axis according to the lasw` x=a sin omegat t`. Find:
`(a)` the mean value of its velocity vector projection `( : v_(x) : )`,
`(b)` the modulus of the mean velocity vector `|( : v : )|`,
(c) the mean value of the velocity modulus `( : v : )` averaged over `3//8` of the period after the start.

Explanation

As `x=a sin omega t` so,` v_(x)=a omega cos omega t`
Thus, `lt v_(x) gt = int v_(x)dt//int dt=(int_(0)^((3)/(8)T)a omegacos(2pi//T)t dt)/((3)/(8)T)=(2sqrt(2)a omega)/(3pi)` (using `T=(2pi)/(omega))`
`(b)` In accordance with the problem
`vec(v)=v_(x)vec(i)`, so,`|lt vec(v) gt|=|lt v_(x) gt|`
Hence, usning part (a) `|lt vec(v) gt|=|(2sqrt(2)a omega)/(3pi)|=(2 sqrt(2)a omega)/(3pi)`
We have got, `v_(x)=a omega cos omegat` So, `{:(v=|v_(x)|=a omega cos omega t, "for", t leT//4),(= -aomega cos omega t,"for", T//4 le t3/8 T):}]`
Hence, `lt v gt =(int f dt )/(int dt)(int _(0)^(T//4)a omega cos omegat dt+int_(T//4)^(3T//4)-a omega cos t dt )/(3T//8)`
Using `omega=2pi//T`, an on evaluating the integral we get
`lt v gt = (2(4-sqrt(2))a omega)/(3pi)`