At what height from the surface of the earth the acceleration of gravity will decrease by 3 times.

Free fall acceleration decreased by 3 times (1/3 g):

1/3 g = G * M / (R + h) ²; g – free fall acceleration on the Earth’s surface (const, g = 9.81 m / s2), G – gravitational constant (const, G = 6.67 * 10 ^ -11 m³ / (kg * s²)), M – mass Earth (const, M = 5.97 * 10 ^ 24 kg), R – Earth radius (const, R = 6371000 m), h – height above the Earth’s surface.

g * (R + h) ² = 3G * M.

(R + h) ² = 3G * M / g.

h = √ (3G * M / g) – R = √ (3 * 6.67 * 10 ^ -11 * 5.97 * 10 ^ 24 / 9.81) – 6371000 = 4664098 m ≈ 4664 km.