Final answer:
The escape speed for a spacecraft from the surface of the Moon is estimated using the ratio of the mass and radius of the Moon to the Earth. Given that the Moon's mass is 1/81 of Earth's and its radius is 0.25 that of Earth's, the escape speed from the Moon is calculated to be approximately 2.38 km/s, which corresponds to the closest given option A: 2.80 km/s.
Step-by-step explanation:
The escape velocity from the surface of an astronomical object like the Earth or the Moon is the speed an object must reach to break free from the object's gravitational influence without further propulsion. The question asks to estimate the escape speed for a spacecraft from the surface of the Moon given that the Moon's mass is 1/81 of Earth's and its radius is 0.25 that of Earth's. Using the formula for escape velocity V = √(2GM/R), where V is the escape velocity, G is the gravitational constant, M is the mass of the celestial body, and R is the radius of the celestial body, we can substitute Earth's escape speed to find the Moon's escape speed. Since mass and radius are given in ratios compared to Earth's, we can simplify the calculation as follows:
- The mass ratio of the Moon to Earth is 1/81, and the gravitational force is proportional to mass,
- The radius ratio of Moon to Earth is 0.25, and gravitational force is inversely proportional to the square of the radius,
- Therefore, the escape speed from the Moon is 0.25 × √(1/81) times that of Earth's escape speed.
The escape speed from the Earth is 11.2 km/s, so the escape speed from the Moon is significantly smaller. When we perform the actual calculation, we find that the escape speed from the Moon is approximately 2.38 km/s, which corresponds to answer option A: 2.80 km/s (as the closest option).