I have tried to realistically model the famous game Agar.io, which can described as the following: A sphere of initial mass $m_0$ expels part of its mass at a given rate ($\frac{dm_l}{dt}$) for thrust with a velocity $v_e$ in the frame of the sphere. At the same time it moves, it is passing through a medium of density $\rho_M$, and absorbs the mass in the volume it has swept/covered, thereby increasing its mass and volume and decreasing its velocity. From this description I want to derive the radius of the sphere as a function of time.
The first thing I noticed is that
$$\frac{dm_G}{dt}=\rho_M \pi r^2v$$
where $m_G$ is the mass gained by absorption. Its rate of change is the volume it has swept per unit of time, multiplied by the medium density.
Also,
$$m=m_0+m_G-m_l$$ $$\frac{dm}{dt}=\frac{dm_G}{dt}-\frac{dm_l}{dt}$$
But since $m=V \rho_s =\frac{4}{3}\pi r^3 \rho_s$, where $\rho_s$ is the density of the sphere,
$$4 \pi r^2 \rho_s \frac{dr}{dt}=\rho_M \pi r^2v-\frac{dm_l}{dt}$$
Another equation to consider, due to conservation of momentum, is
$$mv=(m+dm_G-dm_l)(v+dv)+dm_l(v-v_e)$$
which simplifies to
$$0=(dm_G-dm_l)v+m\cdot dv+dm_l(v-v_e)$$ $$0=v\cdot dm+m\cdot dv+dm_l(v-v_e)$$ $$0=v\frac{dm}{dt}+m\frac{dv}{dt}+\frac{dm_l}{dt}(v-v_e)$$ $$0=4 \pi r^2 \rho_s \frac{dr}{dt}v+\frac{4}{3}\pi r^3 \rho_s\frac{dv}{dt}+\frac{dm_l}{dt}(v-v_e)$$
However, I don't know exactly how to combine these equations into a useful one without the unknown $v$ or $\frac{dv}{dt}$, or higher order derivatives of $r$, since I don't see a way to, for example, get $\frac{d^2r}{dt^2}$ for $t=0$, which doesn't allow me to do a numerical solution.
