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Danger Area Below Cliff Rescue 31-May-2012 At 3:03:34 PM sbm

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I'll have a go at a back-of-the-envelope.

The gravitational energy in a 9kg rock falling 30m is mass * gravitational acceleration * height:

mgh = 9 * 10 * 30 = 2700 kJ.

The splitting into three doesn't really make a difference as you'd also split the kinetic energy into three so the distances are the same.

The rock hitting the ground is a very inelastic collision, lots of energy goes into the boom and the crater and breaking apart the rock. Let me pull a number completely out of my arse and say 50% gets converted into kinetic energy.

1/2 mv^2 = 1350

rearrange and get v = sqrt(2*1350/9) = 17 metres per second.

Maximum range you get from launching a projectile from flat ground at 17 meters per second is

v ^ 2 / g = 17^2 / 10 = 29 meters.

Hey, looks like your rule of thumb came out pretty well!

This also ignores the bouncing and rolling after fragments hit the ground.

Being a bit more formal with the maths, it turns out the maximum distance D a rock could bounce upon hitting the bottom of the cliff, is D = 2 * a * h, where h is the height it fell and a is the percentage of energy converted into kinetic energy in the impact. Assuming a is 50% or 0.5, kierenl's rule of thumb is spot on.