Final answer:
The escape velocity from a spherical asteroid can be calculated using the formula √(2GM/R), where G is the gravitational constant, M is the asteroid's mass, and R is its radius, and then converting to km/s and mph.
Step-by-step explanation:
To calculate the escape velocity from the surface of a spherical asteroid, we can use the formula for escape velocity ve = √(2GM/R), where G is the gravitational constant (6.67430 × 10−12 m3kg−1s−2), M is the mass of the asteroid, and R is the radius of the asteroid. Given the diameter of the asteroid is 10 km (=10,000 m), this makes the radius 5,000 m. The density (ρ) of the asteroid is 3000 kg/m3, so the mass (M) can be calculated using the formula for the volume of a sphere (V = 4/3πR3) and then multiplying the volume by the density.
The steps are as follows:
- Calculate the volume of the asteroid: V = 4/3π(5000 m)3
- Calculate the mass of the asteroid: M = Vρ = 4/3π(5000 m)3 × 3000 kg/m3
- Calculate the escape velocity: ve = √(2 × 6.67430 × 10−12 m3kg−1s−2 × M / 5000 m)
After calculating v
e
, we can then convert the escape velocity from m/s to km/s and then to mph for the complete answer.
To convert from m/s to km/s, simply divide by 1000. To convert from km/s to mph, multiply by the conversion factor of 0.621371. Note that the speed conversion calculations and the final answers should be rounded to maintain two or three significant figures as specified in the question.