A rocket uses a gas of density ro for propulsion, and assume it to be constant for this problem. The pressure of the gas is P inside the rocket with a tank of cross-sectional area A, and the pressure is P0 just as it exits the rocket through a nozzle of cross-sectional area A0.
Use Bernoulli's equation and the equation of continuity to find an expression for the exhaust velocity of the gas, and an expression for the thrust of the rocket.
Work:
Bernoulli's equation:
P + (1.2)*ro*v^2 + ro*g*y = a constant, where P = pressure and ro = density
Continuity equation: R = Av = a constant
P = Mv, where M = mass and v = velocity
Mv = -dmU + (M + dM)*(v + dv), where U = (v + dv - u)
Mv = -dm(v + dv - u) + (M + dM)*(U + dv)
Mv = -dm(v + dv - u) + (M + dM)*(V + dv)
Mv = -dMv - dMdv + dMu + Mv + Mdv + dMv + dMdv
-dMu = Mdv
-(dM/dt)*u = M(dv/dt)
Solution:
Thrust of rocket: Ru = Ma, where -dm/dt = R and a = dv/dt
Exhaust velocity of the gas: U = (v + dv - u) and u = (v + dv) - U