.........about which formula is the correct one to attain the highest energy input into the cue ball.........
Only my two cents
Here not so simple process takes place. There is no direct transfer of kinetic energy of breakcue to kinetic energy of cueball as is present elastic deformations. Shortly so. The law of conservation of energy but as the theory of elastic impact here operates.
In the beginning, kinetic energy of movement of breakcue is transformed to potential energy of deformation of system cueball-tip-shaft-butt. This potential energy collects and given to cueball for very short time of contact of a tip with cueball. This time - shares of second. Therefore the cueball has huge acceleration and cue has huge loadings.
Break shot can be described by means of " the theory of elastic impact ". I shall miss formulas. These formulas and methods of calculations can be found in directories, books and articles. But also formulas are not so important for us as precisely calculate of a cue is impossible. But regularity are important. The basic regularitys is such.
1. The less rigidity of system tip-shaft-butt along grain - the more time of contact but more loss for deformation of system tip-shaft-butt, less acceleration and as a result less speed of cueball after a shot. Therefore nobody use soft tips for break cues for example.
2. The higher rigidity of system tip-shaft-butt - the is less time of contact, but less loss on deformation, acceleration is higher and as a result speed of cueball after a shot is higher. But before the certain repartitions. Try to break with a steel core - any pleasure

. Therefore many use the most rigid tips for breakcues and the most rigid along grain wood. But not the most rigid, for example ebony or ipe. My preference the Hornbeam and Purple Heart. A firm mountain maple is good too.
3. The higher speed of breakcue at the moment of contact with cueball - the higher speed of cueball after a shot. Hence a task of the player to accelerate breakcue till the maximal speed on the minimal way.
Notice, I did not mention about weight of cue. It is important certainly too, but it is present only at calculation of elastic deformation at the moment of contact and has no direct influence on speed of a cueball after a shot. As the weight of cue always is much greater weights of a cueball that of essential influence on final speed of a cueball it does not render. Gaspard-Gustave Coriolis (1792 - 1843) in 19 century has determined weight of cue for billiards. He has proved in the book THEORIE MATHEMATIQUE DES EFFETS DU JEU DE BILLARD (MATHEMATICAL THEORY OF THE EFFECTS OF THE GAME OF BILLIARDS) that the weight of billiard cue should be approximately (if I am not mistaken) 2.4-3.2 weights of a cueball for the most effective transfer of energy, including rotation, to a ball.
Weight of the majority pool cues is from 3.1 up to 3.4 weights of cueball. Weight of breakcues from 2.8 up to 3.2 weights of cueball. The smaller weight of breakcues is subjective parameter and is caused by the physical data of the human (most likely really). To human is easier to accelerate a breakcue up to maximal speed on small distance than cue is lightweight .
Sorry for my bad English. Probably I not have named all terms correctly. Almost lecture has turned out. Probably an echo my former profession

. I am Cand.Tech.Sci., the senior lecturer of faculty "Theoretical mechanics". And now I am cuemaker.
