Answer
The loss of gravitational potential energy of the cabbage–Earth system equals twice the gain in the spring’s potential energy.
Work Step by Step
We can find an expression for the distance $y$ that the spring stretches:
$mg = ky$
$y = \frac{mg}{k}$
We can find an expression for the loss of gravitational potential energy:
$mgy = (mg)(\frac{mg}{k}) = \frac{(mg)^2}{k}$
We can find an expression for the gain in spring potential energy:
$\frac{1}{2}ky^2 = \frac{1}{2}~k~(\frac{mg}{k})^2 = \frac{1}{2}\frac{(mg)^2}{k}$
We can see that the loss of gravitational potential energy of the cabbage–Earth system equals twice the gain in the spring’s potential energy.
