> [!NOTE] Definition > Newton’s law of gravitation yields that the height $y(t)$ then satisfies (until it drops to the ground again) the [[Scalar Ordinary Differential Equation|ode]] $\frac{d^{2}}{dt^{2}} = -\frac{g}{(1+y(t)/R)^{2}}$where $g \approx 9.81 \; ms^{-2}$ is the earth's gravitational constant and $R \approx 6.4 \times 10^{6} m$ is the Earth's radius. > > At launch it satisfies the initial conditions $y(0)=0, \; y'(0)=v.$ # Properties **Scaling**: By [[Scaled Model for Projectile Launched Vertically From The Earth]], ... Solution: By [[Regular Perturbation of Scaled Model for Projectile Launched Vertically From The Earth]], ...