Exercise 1.5: Gravitational force

According to Newton's law of gravitation, the action of the gravitational field of the earth on a particle is given by

$$F=\frac{GMm}{(R+y)^2}=\frac{gm}{(1+y/R)^2},$$

where $y$ is measured from the earth's surface, $R$ is the earth's radius, $G$ is the gravitational constant, $M$ is the mass of the earth, and $g=GM/R^2$. There is not simple analytical solution for this problem. Modify your code to simulate the fall of a particle from an altitude $y_0$ with zero initial velocity, and compute its speed when it hits the ground. Determine the value of $y_0$ for which this impact velocity differs by one percent from its value under a constant acceleration $g=9.8m/s^2$. Take the radius of the earth to be $6.37\times 10^6m$.