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

Options

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

Answer

B

Explanation

Delivering power of a machine
P = constant
`P = FU " "[ :' "Power" = W/t = F cdot S/t = F.v] " " = mav = m(dv)/(dt)v`
`P = m v (dv)/(dt) , " " vdv = P/m dt`
Integrating on both sides, we get
`int_0^v v dv = int_0^t P/m dt , " " (v^2)/(2) = (Pt)/(m) " " implies v = ((2 Pt)/(m))^(1//2)`
`v = (ds)/(dt) = ((2Pt)/(m))^(1//2) implies ds = ((2Pt)/(m))^(1//2) dt " " implies s = sqrt((2P)/(m)) (t^(3//2))/(3//2) " ":. " "s prop t^(3//2)`