A space probe is in an elliptical orbit around the Moon (lunar gravitational parameter $\mu_M = 4{,}902.8\text{ km}^3/\text{s}^2$, lunar radius $R_M = 1{,}737\text{ km}$). The orbit has a perilune radius $r_p = 1{,}850\text{ km}$ and apolune radius $r_a = 5{,}550\text{ km}$. The speed of the probe at perilune $V_p$ (in $\text{km/s}$) is closest to:

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### 1. Strategy & Theoretical Principles From the Vis-Viva energy equation of two-body orbital mechanics: $$V^2 = \mu \left(\frac{2}{r} - \frac{1}{a}\right)$$ where $a$ is the semi-major axis of the elliptical orbit: $$a = \frac{r_p + r_a}{2}$$ ### 2. Analytical Formulation Given: - $r_p = 1{,}850\text{ km}$ - $r_a = 5{,}550\text{ km}$ - $\mu_M = 4{,}902.8\text{ km}^3/\text{s}^2$ Compute semi-major axis $a$: $$a = \frac{1850 + 5550}{2} = \frac{7400}{2} = 3{,}700\text{ km}$$ ### 3. Step-by-Step Calculation Substitute into the Vis-Viva relation at perilune ($r = r_p = 1{,}850\text{ km}$): $$V_p^2 = 4902.8 \times \left(\frac{2}{1850} - \frac{1}{3700}\right)$$ Notice that $\frac{2}{1850} = \frac{4}{3700}$: $$\frac{2}{1850} - \frac{1}{3700} = \frac{4 - 1}{3700} = \frac{3}{3700}$$ $$V_p^2 = 4902.8 \times \frac{3}{3700} = \frac{14{,}708.4}{3700} \approx 3.97524\text{ km}^2/\text{s}^2$$ $$V_p = \sqrt{3.97524} \approx 1.9938\text{ km/s} \approx 1.99\text{ km/s}$$ ### 4. Distractor Forensics - At apolune: $V_a = V_p \frac{r_p}{r_a} = 1.9938 \times \frac{1850}{5550} = 1.9938 / 3 \approx 0.665\text{ km/s}$. - Circular orbit speed at $r_p$: $V_c = \sqrt{4902.8/1850} = 1.628\text{ km/s}$. ### 5. Exam Trap & Standard Reference - **Trap Alert:** Angular momentum is conserved: $h = r_p V_p = r_a V_a$. - **Standard Reference:** Curtis, H.D., *Orbital Mechanics for Engineering Students*, Chapter 2.

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