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PhysicsNEET 2024Kepler's laws, satellites, escape velocity

Gravitation

The minimum energy required to launch a satellite of mass mm from the surface of the earth (mass MM, radius RR) into a circular orbit at an altitude of 2R2R from the surface of the earth is:

  1. A.2GmM3R\dfrac{2GmM}{3R}
  2. B.GmM2R\dfrac{GmM}{2R}
  3. C.GmM3R\dfrac{GmM}{3R}
  4. D.5GmM6R\dfrac{5GmM}{6R}✓

Solution

Orbit radius r=3Rr = 3R. Energy in orbit =−GMm2r=−GMm6R= -\frac{GMm}{2r} = -\frac{GMm}{6R}; on the surface =−GMmR= -\frac{GMm}{R}. Minimum energy =−GMm6R+GMmR=5GMm6R= -\frac{GMm}{6R} + \frac{GMm}{R} = \frac{5GMm}{6R}.

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