8.1.4.19 Geometric and physical applications

Problem: Determine how long it will take a particle, attracted by the Earth, to fall on the Earth, according to Newton's law (with an acceleration, inversly proportional to the square of the distance between them), if at the initial moment the velocity of the particle is equal to zero, and its distance from the centre of the Earth is equal to \( H \). Neglect the atmospheric resistance. The acceleration of the free fall on the Earth's surface is constant and equal to \( g \).