Savchenko Problem 2.6.17

The first cosmic velocity is the minimum speed needed to orbit at a planet's surface—it's where gravitational pull provides exactly the centripetal force to keep you in circular motion. For any body, this velocity depends only on surface gravity and radius: $v_1 = \sqrt{gR}$. The orbital period then follows from how far you travel (the circumference) divided by that speed. Since the Moon is smaller and less massive than Earth, you'll find both its surface gravity and orbital velocity are much lower, leading to a shorter orbital period despite the smaller orbit.

The first cosmic velocity is the minimum speed needed to orbit at a planet's surface—it's where gravitational pull provides exactly the centripetal force to keep you in circular motion. For any body, this velocity depends only on surface gravity and radius: $v_1 = \sqrt{gR}$. The orbital period then follows from how far you travel (the circumference) divided by that speed. Since the Moon is smaller and less massive than Earth, you'll find both its surface gravity and orbital velocity are much lower, leading to a shorter orbital period despite the smaller orbit.