The captain of a spaceship orbiting planet X discovers that to remain in orbit at $410 \mathrm {~km}$ from the planet's center, she needs to maintain a speed of $68 \mathrm {~m} / \mathrm { s }$ What is the mass of planet X? (G = 6.67 × 10^-11 N ∙ m²/kg²) A) 2.8 × 10¹⁹ kg B) 4.2 × 10¹⁷ kg C) 2.8 × 10¹⁶ kg D) 4.2 × 10¹⁴ kg

Question

Quizplus · Accepted Answer

The centripetal force required for circular orbit is provided by gravitational force: $ \frac{mv^2}{r} = \frac{GmM}{r^2} $. Solving for $ M $, the mass of the planet, gives $ M = \frac{v^2r}{G} $. Substituting $ v = 68 \, \mathrm{m/s} $, $ r = 410,000 \, \mathrm{m} $, and $ G = 6.67 \times 10^{-11} \, \mathrm{N \cdot m^2/kg^2} $, we find $ M \approx 2.8 \times 10^{19} \, \mathrm{kg} $.

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The Captain of a Spaceship Orbiting Planet X Discovers That $410 \mathrm {~km}$