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GATE 2025 AE – Question 53

Flight Mechanics & Space Dynamics · Space dynamics: central force motion, Keplerian orbits, Kepler's laws and escape velocity · 2 marks · Numerical answer

F and G denote two points on a spacecraft's orbit around a planet, as indicated in the figure. O is the center of the planet, P is the periapsis, and the angles are as indicated in the figure. If OF = 8000 km, OG = 10000 km, $\theta_F = 0$ deg, and $\theta_G = 60$ deg, the eccentricity of the spacecraft's orbit is ____ (rounded off to two decimal places).

Part of an orbit with the planet's centre O on the left. P is the periapsis on the axis through O. The points F and G lie on the orbit, with the angles $\theta_F$ and $\theta_G$ measured at O from the periapsis direction.

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Correct answer: 0.66 to 0.68

Explanation

The orbit equation is $r = \frac{p}{1 + e\cos\theta}$. At F ($\theta = 0$): $8000 = \frac{p}{1 + e}$. At G ($\theta = 60^\circ$): $10000 = \frac{p}{1 + 0.5e}$. Equating $p$: $8000(1 + e) = 10000(1 + 0.5e)$, so $3000e = 2000$ and $e = 0.67$.