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Problem 4.165 [Difficulty: 3] y x CS at speed U V e Y X Given: Data on rocket weapon Find: Expression for speed of weapon; minimum fraction of mass that must be fuel Solution: Basic equation: Momentum flux in x direction Assumptions: 1) No resistance 2) p e = p atm 3) Uniform flow 4) Use relative velocities 5) Constant mass flow rate From continuity dM dt m rate = constant = so MM 0 m rate t ⋅ − = (Note: Software cannot render a dot!) Hence from momentum a rfx − M ⋅ dU dt − M 0 m rate t ⋅ − () ⋅ = u e ρ e V e ⋅ A e ⋅ ⋅ = V e − m rate ⋅ = Separating variables dU V e m rate ⋅ M 0 m rate t ⋅ − dt ⋅ = Integrating from U = U 0 at t = 0 to U = U at t = t
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