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A rocket is fired vertically from the Earth's surface, directly away from the Sun, along the Sun-Earth line. The Sun's mass is 3 x 10⁵ times the Earth's mass, and the Sun-Earth distance is 2.5 x 10⁴ times the Earth's radius. The escape speed from the Earth alone is vₑ = 11.2 km/s. Ignoring Earth's rotation and revolution and other bodies, the minimum launch speed vₛ for the rocket to escape the entire Sun-Earth system is closest to:
- vₛ = 42 km/s
- vₛ = 22 km/s
- vₛ = 72 km/s
- vₛ = 62 km/s
Correct answer: vₛ = 42 km/s
Solution
vₛ² = (2GM_E/R_E) + (2GM_S/d) = vₑ² + 2GM_S/d. Now 2GM_S/d = (2GM_E/R_E)(M_S/M_E)(R_E/d) = vₑ² (3e5/2.5e4) = 12 vₑ². So vₛ² = 13 vₑ² => vₛ = vₑ sqrt(13) = 11.2 x 3.606 ≈ 40.4 km/s, closest to 42 km/s.
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