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I heard that a good way to test if a battery is dead or alive is to see if it bounces. Supposedly, dead batteries bounce higher than living ones. Can someone tell me if this is a legitimate claim, and if it is, an explanation would be nice.

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Yes.

The battery bounce test kind of works. You can tell whether the battery still has a charge greater than 90%, but bounce hight levels off at about 50% and doesn't go higher. This means that it is not clear whether the battery is completely discharged when you observe a bounce.

Why?

Steingart et al. just recently published a paper on the subject, and through scientific rigor could confirm this claim. The reason they give has to do with how an alkaline battery is built:

Schematic cut through an alkaline battery.

(Source)

As you can see, a large part of the inside is devoted to the zinc anode. During the discharge, a redox reaction takes place to produce the current: $$ \ce{Zn + 2MnO2 -> ZnO + Mn2O3} $$ The zinc (usually in some gel form at first) is slowly oxidized. As a result of this oxidization, crosslinks between the small zinc granules are formed, which make the whole center more rigid. As such, its "bounce" (or coefficient of restitution) becomes greater, because the downward kinetic energy cannot be fully absorbed by the gel anymore.

If you are interested in more details, check out the article they published. If you don't have access you can read a summary given on the Princeton University News web page, where you also can see an animation of a series of discharged batteries bouncing to different heights.

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    $\begingroup$ I have to say, I love how sciencey Steingart et al. sound with their paper title, and how it can be decrypted to just some plain battery bouncing! $\endgroup$ – Nicolau Saker Neto May 21 '15 at 22:38

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