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 mass of planet X can be calculated using the formula for gravitational force $F = \frac{Gm_1m_2}{r^2}$ and the formula for centripetal force $F = \frac{mv^2}{r}$ required to maintain an orbit. Setting these equal to each other because the gravitational force provides the centripetal force needed for orbit, we get $\frac{Gm_1m_2}{r^2} = \frac{mv^2}{r}$. Simplifying and solving for $m_1$ (the mass of the planet), we get $m_1 = \frac{rv^2}{G}$. Plugging in the given values ($r = 410 \, \text{m}$, $v = 68 \, \text{m/s}$, $G = 6.67 \times 10^{-11} \, \text{N} \cdot \text{m}^2/\text{kg}^2$), we find the mass of planet X to be approximately $2.8 \times 10^{19} \, \text{kg}$.

The Captain of a Spaceship Orbiting Planet X Discovers That 410 km410 \mathrm {~km}410 km

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