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GATE 2024 AE – Question 39

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

At a point in the trajectory of an unpowered space vehicle moving about the Earth, the altitude above the mean sea level is 600 km, and the speed with reference to a coordinate system fixed to the center of mass of the Earth is 9 km/s. Assume that the Earth is a sphere with a radius 6400 km and $GM_{Earth} = 3.98 \times 10^{14}$ m$^3$/s$^2$, where, G is the universal gravitational constant and $M_{Earth}$ is mass of the Earth. The trajectory is:

  1. Circular
  2. Elliptic
  3. Parabolic
  4. Hyperbolic

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Show answer and explanation

Correct answer: (B) Elliptic

Explanation

At $r = 7000$ km the circular speed is $\sqrt{\frac{\mu}{r}} = 7.54$ km/s and the escape speed is $\sqrt{\frac{2\mu}{r}} = 10.66$ km/s. The speed of 9 km/s is between them, so the energy is negative and the orbit is an ellipse.