Question Detail
Question
The mean diameters of Mars and Earth are 6.9 × 10³ km and 1.3 × 10⁴ km, respectively. What is the value of the gravitational acceleration on Mars and the escape speed on Mars?
Correct Answer
Option A is the correct answer.
Detailed Explanation
Gravitational acceleration g = GM/R². Using Earth as reference: g_Mars/g_Earth = (M_Mars/M_Earth) × (R_Earth/R_Mars)². Mars mass ≈ 0.107 Earth masses. R_Mars = 3.45 × 10³ km, R_Earth = 6.5 × 10³ km (half of diameters). So g_Mars = 9.8 × 0.107 × (6.5/3.45)² = 9.8 × 0.107 × (1.884)² = 9.8 × 0.107 × 3.55 = 9.8 × 0.38 = 3.72 m/s² ≈ 3.8 m/s². Escape speed v_esc = √(2GM/R) = √(2gR). For Mars: v_esc = √(2 × 3.72 × 3.45 × 10⁶) = √(25.67 × 10⁶) = 5.07 × 10³ m/s = 5.07 km/s ≈ 5.0 km/s.
Hint
Use ratios based on Earth values. g ∝ M/R², escape velocity ∝ √(M/R).