Right before the rope snaps, there’s a ton of tension that is built up from both teams pulling on it; the rope is only stationary because it’s being pulled equally in both directions.
When it snaps, all that tension is instantly released; the other side of the rope is no longer holding this side of the rope in place (and vice versa; this side isn’t holding that side in place), so the rope will whip backwards at a tremendous speed, cutting through basically anything in its way. A heavy duty tow cable snapping can cut a person in half if they happen to be in the way.
Can you actually build up this much force in a regular game of tug of war? Probably not, because that would require both teams to be perfectly evenly matched (otherwise one side would just pull the other down and win) and extremely strong (or you won’t get the tension built up enough to snap the rope).
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