The escape speed from the surface of the Earth is 13.5 × 10³ m s⁻¹. Estimate the escape speed for a spacecraft from the surface of the Moon. The Moon has a mass 1/81 that of Earth and the radius 0.25 that of Earth.
To estimate the escape speed for a spacecraft from the surface of the Moon, you can use the concept of escape velocity.
The escape velocity can be calculated using the formula:
Ve = √(2GM/r),
where:
Ve is the escape velocity,
G is the gravitational constant (approximated as 6.67430 × 10⁻¹¹ m³ kg⁻¹ s⁻²),
M is the mass of the celestial body (in this case, the Moon),
and r is the distance from the center of the celestial body to the object (in this case, the radius of the Moon).
Given that the mass of the Moon is 1/81 of the Earth's mass and the radius of the Moon is 0.25 of the Earth's radius, we can use these values to estimate the escape velocity.
Step 1: Calculate the values for the Moon:
- Mass of the Moon (Mm) = (1/81) * Mass of Earth = (1/81) * M
- Radius of the Moon (rm) = 0.25 * Radius of Earth = 0.25 * r
Step 2: Substitute these values into the escape velocity formula:
Ve = √(2GMm/rm).
Step 3: Simplify the equation using the numerical values:
- G ≈ 6.67430 × 10⁻¹¹ m³ kg⁻¹ s⁻² (gravitational constant)
- M ≈ Mass of Earth (unknown value)
- Mm ≈ (1/81) * M (mass of the Moon in terms of Earth's mass)
- r ≈ Radius of Earth (unknown value)
- rm ≈ 0.25 * r (radius of the Moon in terms of Earth's radius)
Step 4: Plug in the values into the formula and solve:
Ve = √(2 * 6.67430 × 10⁻¹¹ * (1/81) * M / (0.25 * r)).
Step 5: Simplify the equation further and calculate the escape velocity.
By following these steps and plugging in the specific values for the mass and radius of the Moon into the formula, you can estimate the escape speed for a spacecraft from the surface of the Moon.