A rocket of initial mass m, moving with a velocity of v, 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. `((rhogA)/(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 rho`
`:.F=(dm)/(dt).upsilon=Aupsilon rho.upsilon=A upsilon^(2).rho`
Least force required `= m_(0)g`
`:.` `m_(0)g = A.upsilon^(2)rho`
or`upsilon^(2)=(m_(0)g)/(Arho)`
`upsilon=((m_(0)g)/(Arho))^(1)/(2)`
(c) is the choice.