I'm a bit stuck with this situation:
Suppose that an object $A$ with mass M=8 kg at 10 mt above the floor falls from rest tied to an 2 mt elastic rope, which does not exert any force since $A$ falls 2 (and I will model it as a spring with elastic constant $k$). I know it will bounce at a minimum height - doesn't matter where it is, but let it be $y_1$-, and at that moment the object decrease its mass in 3 kg, so that new mass is m=5 kg.
I want to know the maximum height it will reach after bouncing, so I set up conservation of energy (assuming there is no friction due to the air). Now, which is the elongation of the spring? I mean, we measure it respect to the equilibrium position. Initially, but it changes when mass changes (it is a a higher position when $A$ weighs less). Although, if elongation is higher after losing mass, I think it implies system has more potential than before but no force has done work to change it. It seems like system has gained energy, which looks absurd.
So, what elongation should I use when I set up the eq: $$Mgy_1 + \frac{1}{2}k(y_\rm{eq}-y_1)^2 = mgy_2 + \frac{1}{2}k(\hat{y_\rm{eq}}-y_2) ^2 $$
Here some images to illustrate this situation.
Where $y_2$ will be the maximum height after bouncing.
Remember $y_\rm{eq}$ is the equilibrium position when $A$ has mass M, and $\hat{y_\rm{eq}}$ is the same when $A$ has mass m.
I hope you could help me to see it more clear. Please, correct me if I'm wrong.