A rocket of initial mass `m_(0)` moving with a velocity of `upsilon` discharges a jet of gases of mean density `rho` and effective area A. The minimum value of v of fuel gas which enables the rocket to rise vertically above is nearly :

Options

A. `((rhog)/(m_(0)A))^(1//2)`
B. `((rhog)/(m_(0)))^(1//2)`
C. `((m_(0)g)/(rhoA))^(1//2)`
D. `((2m_(0)gA)/(rho))^(1//2)`

Answer

c

Explanation

Here `(dm)/(dt) = Aupsilon`
`:.F=(dm)/(dt)upsilonrho.upsilon=A.upsilon.rho`
Least force required `= m_(0)g`
`:. m_(0)g = A.upsilon^(2)rho`
or `upsilon^(2)=(m_(0)g)/(Arho)`or   `upsilon=((m_(0)g)/(A rho))^(1)/(2)`
(c) is the choice