Hooray

In this Slate.com article, writer Sonia Smith notes that

[a]ccording to the Oxford English Dictionary, the Latin verb explodere means "to drive out by clapping, hiss (a player) off the stage."

That's all well and good, but I could never leave it at that: where does "explodere" come from? It's obvious that "ex-" means "out", as it generally does in Latin, but what about the "-plodere" part? I thought about it for a bit while I was doing the dishes (a great time to think), and then it hit me: "clapping" is "applause", "explodere" means "to drive out with clapping", and therefore "-plodere" must be a variant of "plaudere", "to applaud". And that's exactly the case: "explode" originally meant literally "to clap out", and through the usual series of mutations and imaginative leaps gradually came to mean any sort of loud disturbance or drastic change, whether emotional or physical, literal or metaphorical.

I would have thought that "laud" and the various "plaudere" words such as "plaudit" would be related, since they're so close in meaning and spelling, but that doesn't seem to be the case. The OED doesn't mention it; nor do any of my other sources. ("Laud" is from Latin "laudere", which, again, is practically identical to "plaudere".) It feels as if they ought to be related, but if they are, I don't have any proof of it.