A body is moved along a straight line by a machine delivering constant power . The distance moved by the body is time `t` is proptional to

Options

A. `t^(1//2)`
B. `t^(3//4)`
C. `t^(3//2)`
D. `t^2`

Answer

C

Explanation

Power, `P= Fv`
`= m (dv)/(dt) v " "["As" F= ma = m (dv)/(dt)]`
or ` vdv= (P)/(m) dt`
Integrating it, we have `int_(0)^(v) vdv= int_(0)^(t) (P)/(m) dt`
`rArr (v^2)/(2) = (P)/( m) t" "[becauseP= "constant"]`
`v= sqrt((2Pt)/(m))`
As `v=(dx)/(dt)`
`therefore (dx)/(dt) = sqrt((2 Pt)/(m)` or `dx= sqrt((2P)/(m)) t^(1//2) dt`
Integrating i,, we have `int_(0)^(x) dx= int_(0)^(t) sqrt((2P)/( m)) t^(1//2) dt`
`rArr x= (2)/(3) sqrt((2P)/( m) t^(3//2)` or `x= sqrt((8P)/(9m)) t^(3//2)`
`therefore x prop t^(3//2)`.