Difference in answers when using thrust force and energy conservation

Question

The problem I was trying to solve was this one.

Using the conservation of energy, I obtained that the velocity $v$ equals $$\sqrt{\frac{(2lx-x^2)g}{l-x}}$$

In the left segment, the momentum doesn't change, so the net force is zero. The reaction force on hinge A can be equated to the combined effect of the gravitational force and the thrust force(force due to changing mass) on the left segment of the chain . The thrust force is given by $$ \textbf{v}\frac{dm}{dt}$$

Here, the thrust force equals $k\frac{v^2}{2}$, where $k$ is the linear mass density. The $2$ in the denominator is because $$\frac{dm}{dt} = k\frac{d(x/2)}{dt}=k\frac{v}{2}$$

However, when we solve the problem this way, we arrive at a different answer than we would if approached using energy conservation. (Using energy conservation and $F=\frac{dp}{dt}$ for the system, we would get $R(x)=\frac{kg(2l^2+2lx-3x^2)}{4(l-x)}$)

In many such chain problems, using thrust force and energy conservation gives two different answers. Here's an example of another one.

A chain of length $L$ and mass per unit length $\rho$ is pulled on a horizontal surface. One end of the chain is lifted vertically with constant velocity $v$ by a force $P$. Find $P$ as a function of height $x$, and the work done by the force

My general doubt is: Why is there a discrepancy in answers? Which answer is more correct?

Answer 1

Momentum conservation states that: $$ \frac{dp}{dt} = F. $$

You have two forces here and two components of $\dot p$, since both $m$ and $v$ (of the moving part) changes: $$ m\frac{dv}{dt} + \frac{dm}{dt}v = Mg - R. $$

So when you write force balance in your head, you forget that the system has acceleration, so the forces (including thrust) are not balanced.

Finally, we get $$ R=klg-k\frac{l-x}2\frac{dv}{dt}-\frac{d\left(k\frac{l-x}2\right)}{dt}v. $$

To simplify it, we use $\frac{dx}{dt} = v$ and $\frac{dv}{dt}=\frac{dv}{dx}\frac{dx}{dt}=v\frac{dv}{dx}=\frac12\frac{d(v^2)}{dx}$. I believe, you can do the algebra and find out that the answer is the same.