Friction work depends on path length
The mistake
Treating friction like gravity and using only vertical height change.
Why it feels right
Energy methods often compress a path into endpoints, so friction gets accidentally included.
How to catch it
Two rough paths with the same endpoints lose the same mechanical energy in your solution.
The first principle
Kinetic friction is non-conservative: W_f = -f_k times path length. Longer rough paths dissipate more energy even with the same height change.
worked mini-example
Two rough ramps reach the same height, one longer. Which loses more energy to friction?
Common wrong move
Same, because the height is the same.
Correct
The longer ramp, if f_k is comparable, because friction acts over more distance.
Test the repair EN-07 · transfer
include J
Make a fresh prediction before referring to the worked example.