🎯 Adventure 14 — Activity 1 (Solution)

2) Key Equations and Derivation

The Moon stays in orbit because gravity provides exactly the centripetal force required for circular motion.

Newton’s gravitational force:   F = GMm / R²

Circular motion requirement:   F = mv² / R

Set them equal:

GMm / R² = mv² / R

Cancel m and multiply by R:

GM / R = v²

Therefore:

v = √(GM / R)

3) Compute the Moon’s Orbital Speed

Using G = 6.67 × 10⁻¹¹, M = 5.97 × 10²⁴ kg, and R = 3.84 × 10⁸ m:

v ≈ 1018 m/s

v ≈ 1.02 km/s

4) Compute the Orbital Period

Circumference:

2πR ≈ 2.41 × 10⁹ m

Time = distance / speed:

T ≈ 2.37 × 10⁶ s

Convert to days:

T ≈ 27.4 days

5) Interpretation

Why does the Moon not crash into Earth?
The Moon is constantly falling toward Earth, but its sideways velocity keeps missing Earth.

What if the Moon moved more slowly?
Its orbit would shrink and it could spiral inward.

What if the Moon moved more quickly?
Its orbit would grow larger, and if it became fast enough it could escape Earth completely.

1) The Big Idea

The Moon stays in orbit because gravity provides exactly the centripetal force needed for circular motion.