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NEET PHYSICSMedium

The additional kinetic energy to be provided to a satellite of mass m revolving around a planet of mass M, to transfer it from a circular orbit of radius R₁ to another of radius R₂ (R₂ > R₁) is

A

GmM(1/R₁² - 1/R₂²)

B

GmM(1/R₁ - 1/R₂)

C

2GmM(1/R₁ - 1/R₂)

D

1/2 GmM(1/R₁ - 1/R₂)

Step-by-Step Solution

The total mechanical energy (EE) of a satellite of mass mm revolving in a circular orbit of radius rr around a planet of mass MM is given by E=−GMm2rE = -\frac{GMm}{2r}.

  1. Initial Energy: In the orbit of radius R1R_1, the total energy is E1=−GMm2R1E_1 = -\frac{GMm}{2R_1}.
  2. Final Energy: In the orbit of radius R2R_2, the total energy is E2=−GMm2R2E_2 = -\frac{GMm}{2R_2}.
  3. Energy Supplied: The additional energy required to transfer the satellite is the difference between the final and initial total energies: ΔE=E2−E1=(−GMm2R2)−(−GMm2R1)\Delta E = E_2 - E_1 = \left( -\frac{GMm}{2R_2} \right) - \left( -\frac{GMm}{2R_1} \right) ΔE=GMm2R1−GMm2R2=12GMm(1R1−1R2)\Delta E = \frac{GMm}{2R_1} - \frac{GMm}{2R_2} = \frac{1}{2}GMm \left( \frac{1}{R_1} - \frac{1}{R_2} \right).
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