The velocity of a particle is given by `v=v_(0) sin omegat`, where `v_(0)` is constant and `omega=2pi//T`. Find the average velocity in time interval `t=0 `to `t=T//2.`

Options

A. `2V_(0)`
B. `(V_(0))/(2)`
C. `(V_(0))/(pi)`
D. `(2V_(0))/(pi)`

Answer

D

Explanation

`V=V_(0)sinomegat`
`overline(v)=(int_(0)^(T//2)V_(0)sin(omegat)dt)/(int_(0)^(T//2)dt)`
`=V_(0)(|-cos(omegat)|_(0)^(T//2))/(|t|_(0)^(T//2)=((T)/(2)-0))`
`=(2V_(0))/(omegaT)[{-cos((omegaT)/(2))}-{-cos(0)}]`
`=(2V_(0))/(2pi)(-cospi+1)`
`=(V_(0))/(pi){-(-1)+1}` `(cospi=cos180^(@)=-1)`
`=(2)/(pi)V_(0)`