> We know intuitively that a ball atop a 20ft ladder has twice the potential energy of a ball atop a 10ft ladder.
...no ? dropping something 10 times from 1ft is nowhere near energetic/damaging as once from 10tf
...no ? dropping something 10 times from 1ft is nowhere near energetic/damaging as once from 10tf
Energy is force × distance. For gravitational potential (and, equivalently, the kinetic energy acquired when that is tranformed into kinetic energy by a fall), the force is the force of gravitational attraction between the body and the object it is falling toward (the object’s weight) and the distance is the distance of the (actual or potential, as appropriate) fall.
The nonlinearity you observe in damage is because damage is a complicated outcome that depends on a lot more than aggregate energy involved.
A cup might not break at all in a 1 foot fall and might shatter when dropped from 10. The outcome is binary, which isn't useful as a scale of input energy.
Also how it translates to difference in damage depends on how elastic is the collision. If the balls are made of rubber, things get much less dramatic, and energy exchange ratios more obvious.
I can hit the window many times lightly, with zero damage at the end.
Or I could roll of of those taps into one big energetic swing and break it. There are thresholds below which no damage happens.