A liquid propellant rocket engine injector atomizes fuel through orifices of discharge coefficient $C_d = 0.75$. The fuel density is $\rho_f = 800\text{ kg/m}^3$. If the pressure drop across the injector face is $\Delta p_{\text{inj}} = 1.0\text{ MPa}$ ($1.0 \times 10^6\text{ Pa}$), the velocity of the liquid fuel jet $v_j$ (in $\text{m/s}$) issuing from the orifice is closest to:

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### 1. Strategy & Theoretical Principles From Torricelli's law / Bernoulli equation for orifice flow: $$v_{\text{ideal}} = \sqrt{\frac{2 \Delta p}{\rho}}$$ Accounting for the velocity discharge coefficient $C_v$ (or discharge coefficient $C_d$): $$v_j = C_d \sqrt{\frac{2 \Delta p}{\rho}}$$ ### 2. Analytical Formulation & Calculation Given: - $\Delta p = 1.0 \times 10^6\text{ Pa}$ - $\rho = 800\text{ kg/m}^3$ - $C_d = 0.75$ 1. Ideal jet velocity: $$v_{\text{ideal}} = \sqrt{\frac{2 \times 10^6}{800}} = \sqrt{\frac{2{,}000{,}000}{800}} = \sqrt{2500} = 50.0\text{ m/s}$$ 2. Actual jet velocity: $$v_j = 0.75 \times 50.0 = 37.5\text{ m/s}$$ ### 3. Distractor Forensics - Ideal velocity without $C_d$ gives $50.0\text{ m/s}$. ### 4. Exam Trap & Standard Reference - **Trap Alert:** Always multiply by the discharge coefficient $C_d$. - **Standard Reference:** Sutton, G.P. & Biblarz, O., *Rocket Propulsion Elements*, Chapter 8.

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