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

Options

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

Answer

B

Explanation

Power = `Fv = P` =constant
`m (dv)/(dt) v = P " " [ :. F = ma = m (dv)/(dt)]`
or `vdv = P/m dt`
Integrate both sides, we get `int vdv = P/m int dt`
`(v^2)/(2) = P/m t + C_1`
Where `C_1` is a constant of integration.
At `t = 0, v = 0 , so , C_1 = 0`
`:. (v^2)/2 = (P)/(m) t " or " v = ((2p)/(m)t)^(1//2)`
`:. (ds)/(dt) = ((2 Pt)/(m))^(1//2) " or " ds = ((2Pt)/(m))^(1//2) dt`
Integrate both sides, we get
`int ds = int ((2Pt)/(m))^(1//2) dt , s = 2/4 ((2P)/(m))^(1//2) t^(3//2) + C_2`
where `C_2` is a constant of integration.
At `t = 0 , s = 0,` so `C_2 = 0`
`:. s ((8P)/(9m))^(1//2) t^(3//2) " or " t^(3//2)` .