Hi.

Interestingly, this is the topic of my current assignment for my masters!

Just to answer a small part of your question: a warm up is to increase intramuscular temperature. This is thought to help the Gogli Tendon Organ (GTO).

It appears that an old study (Asmussen e et al 1945) showed that after 10-15mins, intramuscular temperature hit the "magic" 39deg C at which they think the beneficial effects begin. It was at this temperature that peak performance was seen.

Hope it helps