A particle of mass `1 kg` has velocity `vec(v)_(1) = (2t)hat(i)` and another particle of mass `2 kg` has velocity `vec(v)_(2) = (t^(2))hat(j)`
`{:(,"Column I",,"Column II",),((A),"Netforce on centre of mass at 2 s",(p),(20)/(9) unit,),((B),"Velocity of centre of mass at 2 s",(q),sqrt68 " unit",),((C ),"Displacement of centre of mass in 2s",(r ),(sqrt80)/(3) "unit",),(,,(s),"None",):}`

Answer

(A) `rarr` q; (B) `rarr` r; (C ) `rarr` p

Explanation

For `1 kg v_(1) = (2t) hat(j) = 4 hat(i) , a_(1) = 2 hat(i) = 2 hat(i)`
for `2 kg v_(2) = t^(2) hat(j) = 4 hat(j) , a_(2) = 2t hat(j) = 4 hat(j)`
(A) Acceleration of centre of mass `= (m_(1)a_(1) + m_(2)a_(2))/(m_(1) + m_(2))`
`vec(a)_(cm) = (2)/(3) hat(i) + (8)/(3) hat(j) rArr a_(cm) = sqrt((4)/(9) + (64)/(9)) rArr (sqrt68)/(3) m//s^(2)`
`f = ma_(cm) = sqrt68 N`
(B) Velocity of centre of mass
`vec(v)_(cm) = (m_(1)v_(1) + m_(2)v_(2))/(m_(1) + m_(2)) = ((4)/(3)hat(i) + (8)/(3) hat(j)) m//s`
`rArr |vec(v)_(cm) | = sqrt((16)/(9) + (64)/(9)) rArr (sqrt80)/(3)`
(C ) `vec(v)_(cm) = (1(2t) hat(i) + 2(t^(2)hat(j)))/(1 + 2) + (2)/(3) that(i) + (2)/(3) t^(2)hat(j)`
Displacement `= int_(0)^(2) vec(v)_(cm) dt = [(3)/(2)((t^(2))/(2))^(2) hat(i) + (2)/(3) ((t^(3))/(3)) hat(j) ]_(0)^(2) = (4)/(3)hat(i) + (16)/(9) hat(j)`
`rArr | "Dispalcement"| = sqrt(((4)/(3))^(2) + ((16)/(9))^(2)) = (20)/(9)` unit