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When I compress a sponge ball, its potential energy changes. Now after releasing the ball, some of its potential energy [which was added to it during compression] changes to kinetic energy. When the ball comes back to its original position or shape, it stops and there doesn't seem any motion in it. So where does the kinetic energy of the ball go? [ here, I am talking of an ideal ball which is perfectly elastic]

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An ideal perfectly elastic ball would not stop oscillating between it's compressed and extended form. It will behave like an ideal spring, which as stretching and letting go will oscillate back and fourth until something external stops it - the ideal elastic ball is a 3D version of this.

What you describe is therefore not an ideal elastic ball. You are describing a very real sponge ball with soft tissue that will absorb the vibrations and convert them to heat as well as with surrounding air that will be put in motion and that way also absorbs energy gradually. Furthermore due to the porosity of such a sponge, air molecules will be filling up the gaps in the ball and their motion when they are sucked back inside the ball after it was compressed is also work done and thus energy spent.

All in all, the ball as well as it's contained air will heat up slightly every time you compress it and let it expand to initial state.

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    $\begingroup$ You beat me to the answer! Nice answer by the way. $\endgroup$ Commented Jan 18, 2018 at 10:50
  • $\begingroup$ Nice answer! Now I understand this. But what if the ball is not perfectly elastic? What if the ball stops before completely regaining its original shape? Then some of its additional potential energy will remain as it is and the remaining will be converted to kinetic energy. Then what will happen to this kinetic energy? Does the ball now begin oscillating about its new mean position, i.e. the position where the ball still retains some of its added potential energy? $\endgroup$ Commented Jan 18, 2018 at 11:29
  • $\begingroup$ @HermoineGranger If the ball does not fully return to its initial size, then some permanent material deformation has happened. The potential energy is now trapped and not potential any more, since it is not delivered back. That's why cars' front ends are designed to crumble in the case of a crash, since we'll rather have the car absorb the energy than the passengers. But to your last point, yes the new size after permanent deformation will be a new "centre" for any vibration or oscillation that might occur. Bend a steal pole and knuckle it, and you will see it vibrate about its new shape. $\endgroup$
    – Steeven
    Commented Jan 18, 2018 at 12:05

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