The position of a particle moving on a straight line depends on time t as
`x=(t+3)sin (2t)`

Options

A. Its velocity at the initial moment is 6 m/s
B. Its acceleration at the initial is `-8 m//s^(2)`
C. It has velocity of 3 m/s at `t=(pi)/2` sec
D. It has an acceleration of `2 m//s^(2)` at `t=(pi)/4` sec

Answer

A

Explanation

`x=(t+3) sin 2t`
`(dx)/(dt)=v=(t+3) 2 cos 2t+sin 2t`
`(d^(2)x)/(dt^(2)) =a=-4 sin 2t(t+3) +4 cos 2t`
`v_(t=0)=6 m//s`
at =0 =`4 m//s^(2)`
`vt=pi//2 (pi//2+3)2cos pi =-(3+pi//2)`
`t=pi//4 =-4(pi//4+3)`