The orbital velocity required to place a spacecraft in a circular orbit around the moon is how related to the orbital velocity required to place it in a similar circular orbit around the earth?

Introduction / Context:
Orbital velocity is the minimum horizontal speed required for a satellite or spacecraft to remain in a stable circular orbit around a celestial body, such as the earth or the moon. It depends on the mass and radius of the central body. The question asks you to compare the orbital velocity around the moon with that around the earth. This comparison tests your understanding of gravitational strength and orbital mechanics, which are important topics in space science and physics.

Given Data / Assumptions:

• We consider circular orbits close to the surfaces of the earth and the moon for comparison.
• Orbital velocity depends on mass M of the central body and radius r of the orbit.
• The mass of the moon is much smaller than the mass of the earth.
• The radius of the moon is also smaller than the radius of the earth.

Concept / Approach:
For a satellite in a circular orbit of radius r around a body of mass M, the orbital velocity v is given by v = sqrt(G * M / r), where G is the gravitational constant. For the earth, M is large and r is relatively large. For the moon, both M and r are smaller, but the mass difference is much more significant than the radius difference. The result is that the gravitational pull of the moon is weaker, and hence the orbital velocity required around the moon is lower than that around the earth. Therefore, the correct qualitative comparison is that orbital velocity around the moon is less than that around the earth.

Step-by-Step Solution:
Step 1: Recall the formula for orbital velocity: v = sqrt(G * M / r).Step 2: For the earth, call the orbital velocity v_e = sqrt(G * M_e / r_e).Step 3: For the moon, call the orbital velocity v_m = sqrt(G * M_m / r_m).Step 4: Note that M_m is much less than M_e, while r_m is smaller than r_e, but the mass difference dominates.Step 5: Since the gravitational field of the moon is weaker, v_m is less than v_e, so the orbital velocity around the moon is lower than around the earth.

Verification / Alternative check:
Numerically, the approximate first cosmic velocity near the earth's surface is about 7.9 kilometres per second. The corresponding orbital velocity near the moon is significantly smaller, roughly around 1.7 kilometres per second. These values are consistent with the reasoning from the formula and show clearly that the orbital velocity around the moon is less. Therefore, the qualitative statement that the required velocity around the moon is less than that around the earth is correct and matches both theory and real space mission data.

Why Other Options Are Wrong:
Option B, greater than the orbital velocity around the earth, would suggest that the moon's gravity is stronger than the earth's, which is false. Option C, equal to the orbital velocity around the earth, ignores the large difference in mass and size between the two bodies and contradicts the formula. Option D, greater than or equal to, also implies the possibility of equal or greater value, which is not supported by gravitational data. Only option A correctly states that the orbital velocity around the moon is less than that around the earth.

Common Pitfalls:
A common mistake is to assume that a smaller body requires higher speed to stay in orbit because its radius is smaller. However, the dominant factor in gravitational attraction is mass. Another pitfall is ignoring the formula and relying solely on guesswork. To avoid such errors, always recall that orbital velocity depends on both mass and radius through v = sqrt(G * M / r), and for the moon the significantly lower mass ensures a lower orbital velocity compared to the earth.

Final Answer:
Less than the orbital velocity around the earth