The escape velocity of a body from the surface of a celestial body depends on the mass and radius of that body. The formula for calculating the escape velocity (Ve) is given by:
Ve = sqrt(2 * G * M / R)
Where:
G is the gravitational constant.
M is the mass of the celestial body.
R is the radius of the celestial body.
In this case, we are comparing the Earth and the Moon. Let's denote the mass of the Earth as ME, the radius of the Earth as RE, the mass of the Moon as MM, and the radius of the Moon as RM.
Given that the mass and radius of the Earth are 81 times the mass and 4 times the radius of the Moon, we can express these relationships as:
ME = 81 * MM
RE = 4 * RM
Now, let's calculate the escape velocity from the surface of the Earth (VeE) using the known values:
VeE = sqrt(2 * G * ME / RE)
Similarly, we can calculate the escape velocity from the surface of the Moon (VeM) using the known values:
VeM = sqrt(2 * G * MM / RM)
Now, we can express the escape velocity of the Moon in terms of the escape velocity of the Earth:
VeM = sqrt(2 * G * (ME / 81) / (RE / 4))
Next, we can substitute the values of ME and RE in terms of MM and RM using the given relationships:
VeM = sqrt(2 * G * (81 * MM / 81) / (4 * RM / 4))
Simplify further:
VeM = sqrt(2 * G * MM / RM)
Now, we can see that the escape velocity of the Moon is simply the square root of the escape velocity of the Earth, as the mass and radius ratios cancel out:
VeM = sqrt(VeE)
VeM = sqrt(11.2 km/s)
VeM ≈ 3.35 km/s
So, the escape velocity from the surface of the Moon is approximately 3.35 km/s. None of the provided answer choices match this value exactly, but the closest option is B. 2.48 km/s.
