As an alkaline battery is discharged, the anode undergoes oxidation from Zn to ZnO.
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The battery most likely begins to bounce because of displacement of water by solid ZnO bridges between particles of zinc in the gel. These bridges provide less impeding and attenuating paths for pressure waves, in turn making the battery bouncier.
Bhadra, S.; Hertzberg, B. J.; Hsieh, A. G.; Croft, M.; Gallaway, J. W.; Van Tassell, B. J.; Chamoun, M.; Erdonmez, C.; Zhong, Z.; Sholkapper, T.; Steingart, D. A. The relationship between coefficient of restitution and state of charge of zinc alkaline primary LR6 batteries. J. Mater. Chem. A. 2015, 3, 9395–9400.