The departure point is this problem:
A water tank on wheels is moving over an horizontal trail with negligible friction. There is a small opening in one of the walls, at a depth of $h$ below the tank's water level. The cross-section area of the opening is $A$. The initial masses of the tank and the water are $M$ and $m_0$. What is the initial acceleration of the cart?
Can we consider the water at the top of the tank to be stationary? If so, then it is pretty straightforward to find the velocity at which the fluid exits the opening. Then I would guess the acceleration could be estimated by looking at momentum variations. However, this is a varying mass system, so the mass also varies. This ends up being similar to the rocket equation, which involves solving a system of differential equations.
Is there a simpler way to solve this kind of problem (in other words, can you obtain the value of the acceleration without having to solve differential equations)?