Powerful and quick moves are possible only with quick alteration between exaltation and restraint in the organism and good functioning of the nervous and muscular systems. The coordination of movements has to be good between different muscles and also within one muscle. Only this way cooperation between different muscles under different strain is possible. Agonists contract and antagonists are stretched simultaneously, this is a precondition for quick movements. When developing vertical jump, the speed of contractions in muscles is very important. Our muscles may be strong, but we may not be able to jump high. This is where the term explosive strength comes into play! I will try to explain this term as simply as possible. The strength of the muscles along with the speed with which you can apply this strength is explosive strength. For example, if we are doing squats with weights, strength has to be applied constantly during the whole movement. This is a very good exercise for increasing muscular strength (maximum strength), but may not be the best exercise for improving vertical jump, as when jumping we need to apply strength for resistance very quickly and suddenly, to push ourselves as high from the ground as possible. Therefore, when our muscles are strong enough, we need to develop explosive strength.
Now, in order to improve the explosiveness, we need to stimulate our central nervous system, which is directly connected to our muscles. To put it in a more simple way, we need to get our muscles working more quickly. I also will continue to emphasise the importance of good flexibility is, therefore stretch stretch stretch!!! If the capability of the muscles to stretch and relax is not sufficient, the amplitude of the movements is smaller.
Also, a proper warm-up ensures a good state of the muscles and, on the other hand, it affects the speed of directing nervous processes. Therefore, when you are tired, both energetic decrease and over acidity in the muscles increases. In the state of tiredness, it is not possible to develop speed. Therefore it is important to perform sprint and vertical jump trainings right after warm-up.
When jumping, the highest strain in the pushing off phase is on the back side of the thigh, as at that moment it is stretched the most. Therefore that part should be warmed up very thoroughly and stretched before training.
This was a short summary on how the nervous system speed affects in our vertical jump. I hope that this helps.