Energy, deformation and restitution
Every stroke is a transfer of energy. The ball arrives with a speed and a rotation; the player adds their own through the arm, the wrist and the trunk. What leaves after contact is the net result of that inventory once the system — topsheet, sponge, blade, ball — has managed it.
The management happens in three linked phases: deformation, storage and restitution. On receiving the load, each component deforms to very different degrees. The topsheet sinks and is dragged in the direction of the impact. The sponge is crushed vertically, reducing its thickness. The blade flexes barely at all, but enough to transmit vibration to the handle. The ball itself flattens against the surface. Each of those deformations works like a compressed spring storing energy for as long as the load lasts. When the system decompresses, each spring returns what it stored and the ball is fired away. No spring is perfect: part is lost to heat and vibration. The mechanical efficiency of a bat consists of returning much and dissipating little.
That circuit explains the speed of the outgoing ball, but not the spin. To understand rotation another element must be added: tangential friction.
The player almost never strikes perpendicular to the surface of the bat. In a forehand loop, for instance, the bat travels on an ascending path: the ball does not collide head on with the topsheet, it grazes it. During the millisecond of dwell, the surface of the bat moves tangentially with respect to that of the ball. If the rubber is able to grip — through tack, through elasticity, or through both — that displacement drags the ball with it. Prevented from moving sideways, the ball does the only thing it can: it spins. It leaves with the rotation the topsheet has imparted. The greater the friction, the greater the conversion of linear motion into spin. The physics of the topsheet as a spin generator is developed in 5.1; the principle is enough here.
From the above emerge two energy circuits worth separating, even though they coexist in every stroke. The translational circuit — incoming speed, compression, restitution, outgoing speed — is decided above all in the sponge and the blade. The rotational circuit — incoming spin, grip of the topsheet, tangential movement, outgoing spin — is decided above all at the surface. The two operate in parallel and not always with the same efficiency: a bat can be efficient at returning speed and mediocre at generating spin, or the other way round. A large part of choosing equipment consists of deciding how efficiency is shared between the two circuits, a theme that runs through chapters 7 and 11.
Finally, a phenomenon that alters this equation is worth recording: modern tensor rubbers, above a certain force threshold, return more energy than would be proportionally expected. This is what the jargon calls the catapult effect, and it is dealt with in detail in 2.4. For now it is enough to know that the energy balance of a bat is not always linear, and that this non-linearity is one of the engines of the contemporary game.