A partical of mass `m` is driven by a machine that deleveres a constant power `k` watts. If the partical starts from rest the force on the partical at time `t` is

Options

A. `sqrt(mK)t^(-1//2)`
B. `sqrt(2mK)t^(-1//2)`
C. `(1)/(2)sqrt(mK)t^(-1//2)`
D. `sqrt((mK)/(2))t^(-1//2)`

Answer

D

Explanation

Here, power `=K wat t, time =t`
as `P=Fupsilon=maupsilon, a=(dupsilon)/(dt)`
`K=m upsilon(dupsilon)/(dt)` or `upsilon d upsilon=(Kdt)/(m)` …(i)
Integrating , we get
`int upsilond upsilon=int (Kdt)/(m)` or` (upsilon^(2))/(2)=(K)/(m)int dt`
or `(upsilon^(2))/(2)=(K)/(m)t` or ` upsilon=sqrt((2Kt)/(m))`
as `a=(d upsilon)/(dt)=(d)/(dt)(sqrt((2Kt)/(m)))=sqrt((2K)/(m))((1)/(2)t^(-1//2))`
`=(1)/(2)sqrt((2K)/(m))`
As `F=ma = mxx (1)/(2)sqrt((2K)/(mt))`or`F=sqrt((mK)/(2t))`
or `F=sqrt((mK)/(2))t^(-1//2)`