Acceleration of a tank leaking water

[guv]

Homework Statement

A tank stores water on the inside. Initially the total mass of the
tank and water is $M$. A constant horizontal force $F$ to the right is applied on the
tank while water starts leaking out at constant rate $r$ (measured
in kilograms/second). Assume the leaked water is momentarily at
rest with respect to the tank and it's leaked from the left side
of the tank.

Determine the acceleration of the tank as a function of time. Ignore all forms of friction and assume the tank moves on a horizontal surface.

Relevant Equations

$$a = \frac{F}{M - rt}$$
$$\vec F = \frac{d (m \vec v)}{dt} = \frac{d m}{dt} \vec v + m \frac{d \vec v}{dt} = - r \vec v + (M - rt) \vec a $$

I would think the first solution is correct but the provided solution to this problem suggests the 2nd solution. Let me know what you guys think about this. Thanks,

 

[kuruman]

Let me know what you guys think about this.

I think that you might profit by reading this insight article that was written specifically to guide one's thinking about problems of this kind.

 

[haruspex]

I would think the first solution is correct

Quite so.
This can easily be seen by setting $F=0, v_0>0$. According to the second solution, the tank accelerates.
The trouble with $F=\frac{d(mv)}{dt}$ is that it only applies to a closed system of masses. The way it has been used in the second equation is as though the leaked water has carried away no momentum with it, imparting all the momentum it had to the tank and the water remaining in it.