*** GALILEO – ΒULLET / MISSILE THEORY - NEWTON : if you shoot (horizontally and straight to the target) with a gun G3 kneeling or lying a target at 300 meters, given that the bullet runs with 800 m / sec , at 1/3 sec the bullet goes to the center of the target and it has not gone down (having a greater range). (I've done it personally hitting the center of the target with 10 continious bullets). If you leave it from the same small height it takes again 1/3 sec the most to reach the ground. (The same happens to a space without air -air resistance- , except that there the target must be put a little farther or a little closer than 300 meters since the speed changes a little). Thus the Galileo proposal that a body that is pushed horizontally will fall to the ground at the same time with a body that is vertically left to fall at the same time from the same height , is not exactly correct but it is quite statistic and has to do with a rather weak push. Further on the basis of Galileo – Newton , in the fall of bodies, the vertical distance is proportional (analogic) to the square of the time since Galileo let objects to fall to the ground from a ski slope : when the body left the ski slope it traveled a horizontal distance (the shadow of which in the ground is the time of the drop) , and also it traveled a vertical distance (which is the distance between the edge of the ski slope and the ground) and this distance is the space of the fall. However , if we select a different – bigger ski slope with much larger vertical length (in the same distance from the ground as before) we shall see that the horizontal vector-time-shadow will be bigger than before because the body in the ski slope has taken much more speed than before. So the analogy is rather ruined. Therefore we see that the physics of proposals (calculus Principia propositions) of Galileo - Newton (F = G mM / r2 , etc.) is a good and reasonable statistical technical approach that weakens in bigger conditions. THE TRUE MAN IS THE MEASURE OF ALL FLUID THINGS.
*** ΑΒΟUT THE ACCELERATION ¨ further , simply put , Galileo's acceleration a =S/t2 means that '' a body moves with eg 4 m/1sec2 '' (ie 1 sec squared). This means that '' the body in 1 sec is moving at 4 m/sec '' or that '' in the end of 1 sec the body covers 4 m in 1 sec i.e. 0,4 min 0,1 sec as momentum speed''. However (since as we said if we choose another Galileo ski slope the Galileo square analogy in time would not be exactly correct) -since the momentum ιnstantaneous speed is fluid , it could be said that : '' at the end of 1 sec the body covers 0 ,4 m in 0.1 sec at the end of which 0.1 sec the body covers 0,04 m in 0,01 sec '' and the acceleration could be defined as a =S/t3 ie 4m/1sec3 (ie sec cubed and not squared). Thus other equations should also be changed. Eg the equation of Galileo for the ''average speed' of an accelerated body S =(1/2)at2 has to do not only with the acceleration of gravity ''G = 9,8 m / sec2'' but also with any other acceleration even larger , so it also could be cubed instead of squared. (Even if we accept the Galileo square , the truth is that a falling accelarating body does not easily accept a definition of a stable ''average speed'' 5m / sec , because if in the first sec it has traveled 5 m, in the 2nd sec it will have traveled much greater distance , for example at least 20 m. And if we put the cube in the acceleration equation it would be even bigger with obvious consequences in the Newtonian ''F = GmM/r2''. So we are talking about rather fluid and calculus type equations that do not involve much wider and bigger conditions.
Trully if you accept ''average speed'' 5m/sec = in 1 sec 5 m , you cannot easilly accept ''average speed'' 20m/2 sec=10m/sec=in 1 sec 10 m , so it is all statistic and fluid.
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In this post i mean 9,8 = g and not G
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