can't quite get my head round this.
Sticking with coming down a hill, assuming for simplicity that you are starting from a stop to look at the view at the top, and that you need to stop at the bottom for a road junction, and imagine that it is a continuous, single, gradient:
- the amount of energy that has to be dissipated is the potential energy resulting from height difference, (mass x gravity x height);
- that energy gets dissipated in various resistances (air drag, bearing losses, tyre hysteresis etc), and in the brakes;
- if you freewheel (no pedalling for the moment please, because that just complicates matters), build-up lots of speed and lots of kinetic energy, then brake from a high speed to a stop at the junction, all the energy being dealt with by the brakes has to be dissipated in a short time, heating-up the braking system as it struggles to shed energy at the rate asked of it;
- if you use the brakes gently, all the way down, speed, and therefore kinetic energy, doesn’t build-up, the potential energy that you had at the start is dissipated at a low rate, dribbled out into the atmosphere by a braking system that can easily shed it so doesn’t heat-up much in the process.
- if you make yourself as “non-aero” as you can, a greater proportion of the energy is dissipated in air drag, so the brakes don’t have to dissipate so much.
- if you are a speed-merchant, and decide not only to defer braking for as long as possible, but to select a high gear and pedal down the hill, you are adding more energy, from your personal reserves, on top of the potential energy resulting from height difference, so giving the brakes yet more to dissipate.
- if you ride over the crest of the hill, rather than starting from a break to look at the scenery, the kinetic energy you have at that moment also has to be got rid of before you can stop at the bottom, so the faster you’re going when you start the descent, the more energy the brakes have to dissipate.
It is all very case/hill dependant though, because there are doubtless some really steep hills where even with the slow-speed, near-continuous gentle braking technique the rate of energy dissipation required is enough to tax some braking systems. Our local short, sharp descent is 65m in about 500m, ending at a T-junction, which isn’t all that evil by some standards, but a quick calculation suggests that coming down it stop-to-stop dissipates enough energy in what feels like next to no time to boil a cup of water.