[bawana]

but this is not intuitive: 'We know intuitively that a ball atop a 20ft ladder has twice the potential energy of a ball atop a 10ft ladder. '

gravity will accelerate a ball. this is not a linear process. the heat generated by collision with the ground is not double, but quadruple.

so the only thing that is linear is the DISTANCE.

Define (a)work = energy, (b)work = force x distance and (c)force = mass x accel. Substitute c into b you get work = distance x mass x accel and substitue into a you get energy = distance x mass x accel.

By this equation, an apple falling twice the distance, (and having a constant mass and acceleration) will only have twice the energy.

This 'lie' of quadratic energy growth is just another magic trick physicists have used to confuse students.