Instead of finding out how much gravitational potential energy PEgrav an object has, often we are more concerned with its initial-to-final change ∆PEgrav, which is the final amount of gravitational potential energy minus the initial amount of gravitational potential energy. Notice how the common factors of m and g (which are presumed to be constant) are pulled out, leaving a "difference of elevations" for the final and initial values of y in the parenthesis.
Instead of finding out how much elastic potential energy PEelas an object has, often we are more concerned with its initial-to-final change ∆PEelas, which is the final amount of elastic potential energy minus the initial amount of elastic potential energy. Notice how the common factors of (1/2) and elastic/spring constant k (which is presumed to be constant) are pulled out, leaving a "difference of squares" for the final and initial values of x in the parenthesis.
In contrast, when non-conservative forces are exerted against the motion of objects, energy is lost, irreversibly converted to non-mechanical forms; or when non-conservative forces work along the motion of objects, energy is gained, but this is from the irreversible conversion of non-mechanical forms.
Here we have many examples of non-conservative forces acting on the cat. As it slides across the floor, friction and drag act against the direction of motion of the cat, removing its translational kinetic energy, and it eventually slows to a stop. This energy is irreversibly lost, dissipated to heating up the cat, contact surfaces, and stirring up the air in the room, and cannot be recovered.
Also the people at either end of the kitchen push on the cat along its direction of motion, adding to its translational kinetic energy. This energy is irreversibly converted from the biochemical reactions in the people, and cannot be recovered.
Because these non-conservative forces involve irreversible transfers to/from mechanical energy forms, we must explicitly account for the non-conservative work done by them in energy conservation accounting.
Note that in the idealized case that there are no non-conservative forces (such as friction, drag, or from external sources), then the left-hand size of this equation would be zero. Then the individual energy terms on the right-hand side of this equation can then trade and balance amongst themselves, instead of with the outside world.
So now let's see how this transfer/balance equation can be applied to idealized situations where friction and drag are negligible.
From just after being launched, to reaching her highest height, as the woman is moving upwards her speed is decreasing, and since translational kinetic energy depends on the square of her speed, as her speed decreases, her translational kinetic energy decreases. (If you calculated her change in translational kinetic energy you would get a negative value, which is consistent with a decrease.)
From just after being launched, to just before reaching her highest height, as the woman is moving upwards her height is increasing, and since gravitational potential energy depends on height, as her height increases, her gravitational potential energy increases. (If you calculated her change in gravitational potential energy you would get a positive value, which is consistent with an increase.)
Since we are assuming friction and drag are negligible, then there is no non-conservative work done on the woman, so the left-hand side of the transfer/balance equation is zero:
Wtr = ∆KEtr + ∆PEgrav + ∆PEelas,
0 = ∆KEtr + ∆PEgrav + ∆PEelas,
and since there are no changes in elastic potential energy while the woman is already moving upwards in the air, we are only left with the changes in the translational kinetic energy and gravitational potential energy terms:
0 = ∆KEtr + ∆PEgrav + 0,
0 = ∆KEtr + ∆PEgrav,
0 = (+) + (–).
Now we can see that amount that the woman's translational kinetic energy decreases (where ∆KEtr is negative) is directly related to the amount that the woman's gravitational potential energy increases (where ∆PEgrav is positive), in order to equal the zero on the left-hand side of the equation. So translational kinetic energy "feeds" (or is "transferred to") gravitational potential energy during this process.
Assuming that there is no friction/drag (such that we can neglect non-conservative work), is energy being transferred from translational kinetic energy to elastic potential energy, or is it transferred the other way around? (Video link: "The Physics of Slingshots 2 | Smarter Every Day 57.")
Assuming that there is no friction/drag (such that we can neglect non-conservative work), which energy system is "feeding" the others? Which energy system experienced a greater amount of change (increase or decrease): translational kinetic energy, gravitational potential energy or the elastic potential energy? (Video link: "Homemade Bungee jump.")