The acceleration of a particle is given by `vec(a) = [2hat(i)+6that(j)+(2pi^(2))/(9) "cos" (pit)/3hat(k)]ms^(-2)`
At `t=0 ,vec(r)=0 and vec(v) =(2hat(i)+hat(j))ms^(-1)` The position vector at `t=2` s is

Options

A. `(8hat(i)+10hat(j)+hat(k))m`
B. `(8hat(i)+10hat(j)+3hat(k))` m
C. `(3hat(i)+8hat(j)+10hat(k))m`
D. `(10hat(i)+3hat(j)+8hat(k))m`

Answer

B

Explanation

`vec(a)=[2hat(i)+6that(j)+(2pi^(2))/9 cos ((pit)/3) hat(k)] m//s^(2)`
at `t=0, vec(r)=0` and `vec(v)=(2hat(i)+hat(j)) m//s` The position vector at `t=2` sec is
`(dvec(v))/(dt)=[2hat(i)+6that(j)+(2pi^(2))/9 "cos" ((pit)/3) hat(k)]`
`underset(2hat(i)+hat(j)) oversetvec(v)int dvec(v)=underset0overset t int (2hat(i)+6that(j)+(2pi^(2))/9 "cos"((pit)/3)hat(k))dt`
`vec(v)-(2hat(i)+hat(j))=2that(i)+3t^(2)hat(j)+(2pi^(2))/9xx3/pi "sin" (pi/3)t hat(k)`
`vec(v)=(2hat(i)+hat(j))+2that(i)+3t^(2)hat(j)+2/3 pi "sin"(pi/3)that(k)`
`vec(v)=[(2+2t)hat(i)+(3t^(2)+1)hat(j)+(2pi)/3 "sin" (pi/3t)hat(k)]`
`vec(v)=(dvec(r))/(dt)=[(2+2t)hat(i)+(3t^(2)+1)hat(j)+(2pi)/3 "sin" (pi/3t)hat(k)]`
`underset0 oversetvec(r) int dvec(r)=underset0overset t int [(2+2t)hat(i)+(3t^(2)+1)hat(j)+((2pi)/3 sin(pi/3t)hat(k))dt`
`vec(r)=(2t+t^(2))hat(i)+(t^(3)+t)hat(j)+(2pi)/3xx(-cos(pit//3))/(pi//3)hat(k)`
`vec(r)=(2t+t^(2))hat(i)+(t^(3)+t)hat(j)-2"cos" (pit)/3 hat(k)`
`vec(r)=(2xx2+4)hat(i)+(8+2)hat(j)-2[cos((2pi)/3)-1]hat(k)`
`vec(r)=8hat(i)+10 hat(j)+3 hat(k)`