In reply to Shearwater:
> This isn't about acceleration due to gravity, it is about acceleration due to reaching the end of your rope and stopping. That'll be a factor of how stretchy the rope is, and the mass of the tethered object.
> Given the same rope, a lighter weight will cause it to stretch less, and so the weight will be brought to a halt in a shorter period of time. Assuming the heavy weight and light weight are falling at the same speed, this implies the lighter weight will experience a greater acceleration.
Shearwater's explanation is exactly right. The acceleration here is a balance of gravity (constant for all masses) and rope stretch (inversely proportional to mass, or something like that).
In the case of the ant -- yes, an ant attached to a climbing rope would hardly stretch the rope at all, and so, yes, it would be brought to a stop almost immediately. Would this fall be dangerous to the ant? Our intuition says no, and that might be right, but it's actually because insects are known to routinely endure accelerations of tens to hundreds of g, where free fall is 1 g (see Burrows and Wolf, J Exp Biol 205, 2002).
But obviously this has no bearing on how dynamic the belay is, or indeed whether a child can put in adequate protection with good judgment, etc. Since most (?) lead-climbing kids weigh at least 60 lbs (I'd guess, from what I've seen) and I weigh about twice that, it's equivalent in terms of acceleration to them taking about four-fold larger falls than me over equivalent fall distance. My intuition says that's fine, but I defer to the judgment of others more knowledgeable than me (and the finer details of any child-lead-climbing situation) on this point.