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GATE 2026 AE – Question 36

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

An earth satellite has the instantaneous position vector $\vec r$ and velocity vector $\vec v$ as given below. Here $\hat p$ and $\hat q$ denote the unit vectors along the $x$ and $y$ axes of the perifocal frame, respectively. Assume that the value of gravitational parameter is 398600 km$^3$/s$^2$. Which one of the following trajectories does the satellite follow?

$\vec r = (8000\hat p + 9000\hat q)$ km and $\vec v = (-6\hat p + 6\hat q)$ km/s

  1. Circle
  2. Hyperbola
  3. Parabola
  4. Straight line

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Correct answer: (B) Hyperbola

Explanation

The specific energy is $\varepsilon = \frac{v^2}{2} - \frac{\mu}{r}$. Here $r = \sqrt{8000^2 + 9000^2} = 12042$ km and $v^2 = 72$ km$^2$/s$^2$, so $\varepsilon = 36 - \frac{398600}{12042} = 36 - 33.10 = 2.9 > 0$. A positive energy means an unbound, hyperbolic orbit.