Where does the energy go?
In basic physics we learn that energy is conserved: it never disappears, it only changes form. We also learn that a mass high up has more energy than the same mass lower down. Today as I was walking down stairs I asked myself, “Self, where is that energy going?”
It seems pretty clear that the energy isn’t going into something moving quickly nor being stored for later use. That means most of it is probably becoming heat.
As a crude rule of thumb, when two things trade kinetic energy for heat most of the heat is absorbed by the less rigid thing. In this case, that would mean my legs, not the stairs. Meaning most of my previous gravitational potential energy ends up as excess heat in my legs. How much heat is that?
There is in the neighborhood of 80kg of me. That means for every meter I descend I’m absorbing 80kg × 9.8 m/s2 = 784 joules of energy. How much heat is that? Water has a specific heat capacity of 4.1813 J/g K meaning my 784 joules is enough to heat about 784 J ÷ 4.1813 J/g K = just under 190 grams of water 1 degree kelvin. Let’s approximate by saying my legs are half my mass and mostly water, meaning each meter adds less than 0.005 K of heat to my legs.
For the buildings I descend, that’s not even noticeable. However, there are about a dozen buildings over 400 meters tall; if I walked down their entire flight of stairs my legs would likely heat up a full two degrees kelvin.