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Using Kepler’s third law: T2 = 4π2GMsa3. Rearrange for Ms: Ms = 4π2a3GT2. T = 2×3.156×107 = 6.312×107s. a = 2.25×1011m. T2 = (6.312×107)2 = 3.984×1015s2. a3 = (2.25×1011)3 = 1.139×1033m3. Ms = 4×(3.14)2×1.139×10336.67×10−11×3.984×1015. Ms = 4.49×10342.657×105≈1.69×1030kg.
Ref: Halliday & Resnick, Fundamentals of Physics, 11th ed., Chapter 13: Gravitation.
In an elliptical orbit, total energy is constant, but kinetic energy (KE = 12mv2) varies as speed changes with distance from the center (faster closer, slower farther), balancing the changing potential energy.
Ref: Halliday & Resnick, Fundamentals of Physics, 11th ed., Chapter 13: Gravitation.
Gravity is a conservative force, so the total mechanical energy (E = KE+PE) is conserved in an elliptical orbit, with kinetic and potential energies trading off but summing to a constant.
Ref: Halliday & Resnick, Fundamentals of Physics, 11th ed., Chapter 13: Gravitation.
In an elliptical orbit, speed varies (faster at perihelion, slower at aphelion) due to conservation of angular momentum, making option 3 correct. As per NCERT, applying relevant law/formula with correct units and sign convention leads to Its speed varies with distance. This satisfies dimensional consistency and physical conditions given, so option C is scientifically correct.
Ref: Halliday & Resnick, Fundamentals of Physics, 11th ed., Chapter 13: Gravitation.
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