What about if it's both mental and physical fatigue which are taking place? Are they really happening in a separate ways? How could it be determined that it's only one of them? Am I wrong or in both cases one have the tendency to fall asleep.

From Fatigue (medical) - Wikipedia

.../Fatigue and ‘feelings of fatigue’ are often confused/...

.../Mental fatigue has also been shown to decrease physical performance/...

.../Both [Compared with sleepiness] are often used interchangeably and even categorized under the description of 'being tired.'/...

It looks like perhaps we are not talking under the same parameters here.