A variable mass problem with friction

[Eitan Levy]

Homework Statement

A water tank with a total mass of m₀ is moving on a horizontal road with a coefficient of friction equals to μ.
At t=0 water starts to come out of the tank with a velocity equal to u₀ in relation to the tank. Each second the mass of water that comes out is λ.

Find the equation of movement of the tank.

The answer is below.

Homework Equations

F=dP/dt

The Attempt at a Solution

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What is my mistake here? What am I doing wrong?

 

[haruspex]

You have to be very careful using $\dot p= \dot mv+m\dot v$. In fact, I wouid never use it.
The trouble is that it treats mass as something which can simply change, independently of anything else. In reality, the mass is going to somewhere or coming from somewhere. The equation works provided that the mass being gained or lost neither brings momentum with it nor takes momentum away with it. That is not the case here.

 

[TSny]

[EDIT: I see that @haruspex posted just before me. I'll leave my comments in case they are helpful.]

$\lambda u_0 = \frac{dm}{dt}u_0$ is the force which an amount of water $dm$ exerts on the "rest of the tank" when $dm$ is ejected during $dt$.

If $m$ is the mass of the tank just before $dm$ is ejected, then the mass of the "rest of the tank" is $m-dm$. This mass of the rest of the tank does not change while $dm$ is ejected. So, when setting up $F_{\rm net} = \frac{dP}{dt}$ for the rest of the tank, you must treat $m-dm$ as constant. So, $\frac{dP}{dt} = (m-dm) \frac{dv}{dt}$. But $dm$ is negligible relative to $m$. So, $\frac{dP}{dt} = m \frac{dv}{dt}$. You had an extra term $\frac{dm}{dt}v$ in $\frac{dP}{dt}$ which shouldn't be there.

Generally, mistakes in dealing with variable-mass problems come from not being careful in defining "the system" and "environment".

The standard derivation of the "rocket equation" found in textbooks avoids having to consider the force which part of the fuel exerts on the rest of the rocket. You just use the momentum-impulse theorem as applied to the entire rocket-fuel system. So, you might want to look up a derivation of the rocket equation in your textbook or elsewhere.