A body of mass `1 kg` begins to move under the action of a time dependent force `vec F = (2 t hat I + 3 t^(2) hat j) N`, where `hat i` and `hat j` are unit vectors along x-and y-axes. What power will be developed by the force at the time `t` ?

Options

A. `(2t^(2)+4t^(4))W`
B. `(2t^(3)+3t^(4))W`
C. `(2t^(3)+3t^(5))W`
D. `(2t+3t^(3))W`

Answer

c

Explanation

`(c )` According to question, a body of mass 1 kg begins to move under the action of time dependent force,
`F=(2thati+3t^(2)hatj)N`
Where` hati "and" hatj` are unit vectors along X and Y-axes.
`thereforeF=ma`
`implies a=F/m`
`impliesa=((2thati+3^(2)thatj))/(1)" "(therefore m=1kg)`
`impliesa=(2thati+3^(2)thatj)m//s^(2)`
`therefore` accelaration, `a=(dv)/(dt)`
`implies" "dv= a dt`
Integrating both sides, we get
`intdv=inta" "dt=int(2thati+3t hatj)dt`
`v=t^(2)hati+t^(3)hatj`
`therefore` Power developed by the force at the time t will be given as
`P=F.v=(2thati+3thatj).(t^(2)hati+t^(3)hatj)`
`=(2t.t^(2)+3t^(2).t^(3))`
`P=(2t^(3)+3t^(5))W`