5x Power Push-ups and 5x Power Jump Squat OTM
Strength, Power and Speed Training
These three type of training utilize the same number of sets and repetition in an exercise; 5 repetition per set or less.
Training Load
What separates them is the Percentage of 1 Repetition Max used in the exercise
1) Strength: 85% plus.
2) Power: 48-62%
3) Speed: 10-40%, with around 30% the sweet spot.
Going Ballistic
Going Ballistic means that a body or object need to be projected into the air; that it goes airborne.
Doing so, ensure that the muscle fibers are worked/trained and developed through the full range of the movement.
Obviously, the load is not only the determinate factor in which type of strength is being used and developed but how high or far the body or and object is projected.
With lighter loaded Speed Training, a greater distance is achieved.
Due to the heavier loading, less distance occurs with Power Training.
With loads closer to 1 Repetition Max, a body or object is not going airborne.
With that said, what is important in trying to go airborne with let's say a Squat is "The Intent".
Compensatory Acceleration Training, Dr Fred Hatfield
Each repetition in a Maximum Strength Set needs to be preformed with "The Intent" to go airborne.
Doing so, innervates and develops the Fast Twitch Muscle Fiber (Type IIa and IIb/X); a muscle fiber that is trained is developed.
Thus, The Intent ensure greater numbers of Fast Twitch are developed.
Body Weight Training
Since there isn't much loading with Body Weight Training, it leans more toward the side of Speed Training.
Take Away
1) Push Ups executed so that you come off the ground, as you do with Jump Squats.
2) Lower Repetition ensure greater Strength, Power and Speed are developed.
As Pet stated, more rest and recovery produce greater power output development.
Like Peter mentioned, once fatigue occurs, power drop. The Fast Twitch Muscle Fiber are no longer being trained nor developed.