I am trying to calculate out something, but no matter how much I try I keep running into issues, such as making formulae that are infinitely recursive without finding an answer.
So I am trying to calculate out the velocity that an object would be at, given drag forces, if it somehow changes mass. I know how to find the velocity ignoring Drag, given Conservation of Momentum.
$m_1v_1 = \frac{1}{am_1v_1}$
$v_2 = v_1 * a$
Kinetic energy would be,
$\frac{(mv_1^2 * a)}{2}$
But I can not find a formula wherein this takes into account how such an object, while speeding up, is also slowed down by drag. To find the true velocity that it would be made to reach.
Note, this is not about a falling object, I know how to do that. This is about an object in a horizontal direction. Furthermore, I know this is better suited towards physics stack exchange, but they are both very picky with what is asked there, and I also feel that since this is mainly based on mathematics, it is better here.
This has been driving me insane and help would be greatly appreciated.
Let $m$ be the mass of the object at time $t$, $v$ be the velocity at time $t$, $F_D$ be the size of the drag force at time $t$. These are all functions of $t$.
Since the only force exerted to the object is drag, we have the following by Newton's second law of motion:
$$\frac{d(mv)}{dt}=-F_D$$ $$\implies v\frac{dm}{dt}+m\frac{dv}{dt}=-F_D$$
Let $\frac{dm}{dt}/m=r$. If we want to find $v$:
$$\frac{dv}{dt}+rv=\frac{-F_D}{m}$$
Apply the formula of the solution to a linear differential equation.
$$v=e^{-\int rdt}\bigg(\int\frac{-F_D}{m}e^{\int rdt}dt \;+C \bigg)$$
All the indefinite integrals have no integration constant and $C$ is a constant. We can use this to find the velocity when $m$ decreased to $\frac{1}{a}m$. Notice the result would be different when how $m$ changes is different.