How (and why) is energy due to friction independent of speed? Consider a rotating disk and a fixed part that "abrades" something. Under normal conditions the disk and the fixed part should not touch. However, they slightly touch due to misalignment. When the disk rotates, this results in friction which ultimately causes heat.
Very intuitively, the higher the angular velocity, the higher the heat generated. A match only ignites when it is moved fast, an angular grinder causes heat when it is fast, moving my hand over a carpet feels only hot/hurts if I do it fast, etc.
The thermal energy released is given by the path integral over the friction force $F$. And the friction force is directly proportional to the normal force $F_N$. The proportionality is a constant ($F=\mu F_N$) and in my case, I think, the normal force is also constant ($F_N=p/A$ where $p$ is constant pressure between the disk and the fixed part and $A$ the contact area). And "Coulomb's Law of Friction" states that the friction is independent of sliding velocity. How can this be?
My actual problem at hand is: Consider the disk as the brake disk of a car and the fixed part the brake jaw (or similar). Assuming I need to drive the car a distance $x$, I would like to find the optimum speed (which is directly related to the angular velocity of the disk) that causes minimum harm. I assume minimum harm is achieved when the heat due to friction is lowest. Intuitively, it is better to drive the car slower but have the friction applied longer than driving fast but have the friction applied for a shorter time. I would like to see this in an equation.
 A: Good question! The answer is that temperature rise is not the same thing as generated heat energy. You would be forgiven for thinking so since
$$ \Delta T = \frac{\Delta E}{C},$$
Where $\Delta T$ is the temperature change, $\Delta E$ is the heat energy change, and C is the heat capacity.
But here’s the thing: $\Delta E$ is not just the added heat energy from all resistive processes over time, it’s the net  energy change at any given time. What’s the difference? Well, every system is connected to the environment in varying degrees. Excess heat energy is constantly being redistributed out to the environment through thermal conduction.
So to increase the temperature, you need to add energy faster than it is subtracted. How does that happen? By converting the motion into heat energy in a short amount of time! Sliding your hands together quickly or slowly will generate the same total energy. But sliding them quickly will generate that energy faster than it dissipates away, giving a noticeable temperature rise. 
