I need some help for a school assignment. I'm trying to make a differential equation to solve in Comsol. The part I'm having trouble with is the fact that air resistance and gravity changes based on the bullets vertical position. I have absolutely no idea how to do this and without it the bullet would always be equally attracted to earth. With the only exception being the drag decreasing with the bullets velocity. If you know of any way to do this, it would be greatly appreciated.
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There is a fundamental problem with this exercise: in the absence of an atmosphere, shooting a bullet upward, when the bullet is (say) a hundred km above the surface it either has escape velocity (in which case it's going to escape) or it doesn't (in which case it is in an elliptical orbit that hits the surface of the Earth). Adding drag doesn't help. So unless you meant a hyperbolic orbit where the bullet just flies out into space farther and farther from the Earth and never actually goes around the Earth, I don't think this works. – David K May 04 '22 at 13:11
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Gravity: $g\approx GM/(R+h)^2$, where $h$ is distance from earth surface. One can solve for atmospheric pressure $p(h)$ with assumption of constant temperature and chemical composition: $\frac{dp}{dh}=-\frac{Mp)}{RT}g$. Then one can use formula for air viscosity and density from $p$ and $T$. Then one can use drag force formula $\vec{F_d}=-C(\mathrm{Re})\rho S \frac{v}2 \vec{v}$ and solve system of ODE: $m\frac{d\vec{v}}{dt}=m\vec{g}+\vec{F_d}$, $\frac{d\vec{r}}{dt}=\vec{v}$, (one needs $\vec{r}$ to find $h(t)$). – Ivan Kaznacheyeu May 04 '22 at 13:54