Uccello: Detail from the Battle of San Romano.
Often, in mounted combat all or much of the energy of a blow is supplied by the horse, a much stronger creature than the man riding it. This is what made a mounted lance so dangerous.
The Bombadier, an early 19th century military manual estimates that a military horse gallops at almost 14 miles an hour, and so a pair of riders charging each other at the gallop have a closing speed of 27 mph.
Under these conditions a lance tip that did not glance could sometimes penetrate the front of a helmet or breastplate, generally among the thickest parts of a harness. For example, in two challenges fought at Arras in 1423, Monstrelet reports that when Poton de Saintrailles and Lyonnel de Wandonne jousted with sharp lances:
They ran many strokes very rudely, and there were several lances broken and crumpled on both sides. Finally, it was seen that the helmet of Lyonnel was somewhat broken by the point of the lance of his adversary, and his head wounded, but not seriously.
Two days later:
The next Rifflard de Champremy, attached to King Charles, and the bastard de Rosebecque ran together with sharp lances. They broke many lances, but, in the end, Rifflard was pierced through his armor, so that one could see his side, but nonetheless was not pierced to the quick. With this blow the duke made them stop; and each party retired to his lodgings with his people.
In each case the armor did its job of preventing serious injury. Against lighter armor elsewhere on the body the lance was more dangerous as in this encounter at St. Inglevert in 1390
The English knight hit sir Reginald a very severe blow on the top of his helmet, without otherwise damaging him; but sir Reginald gave him so strong a thrust on the target, (for at that time he was counted one of the stoutest tilters in France, and was smitten with love for a young lady that made all his affairs prosper) it pierced through it as well as his left arm: the spear broke as it entered, the butt end falling to the ground, the other sticking in the shield, and the steel in the arm.
Swords can be braced against the body and used like a short lance, as in the detail from Uccello's painting of the Battle of San Romano above. John Cruso, writing in 1632, recommends placing the pommel of the curiassier's sword against his right thigh:
...and so with his right hand to direct or raise the point to his mark, higher or lower as occaision serveth: either at the bellie of the adverse horse-man (about the pummmel of his sadddle) or at his arm pits, or his throat...
Pietro Monte, writing in 1509 and Ghislerio in 1587 recommend a position with the sword hand resting on or in front of the front of the saddle-bow. This would have allowed the rider to set the pommel or cross of his hilt against his saddle-bow, transferring the force of the blow's impact to his horse through the saddle.
Edge blows could also use the mount's momentum to strengthen a blow, combining that energy with force supplied by the rider's body, shoulder and arm. Duarte, King of Portugal, writing in the 1430s, stressed the importance of using the strength of horse and man together:
To give a forceful cut it must be made as the horse approaches and with the body and arm acting together. I find this very useful in the tourney, because when one strikes when at a standstill and with the arm alone the blow is comparatively weak, whereas if the horse is moving and the body and arm together it is far stronger.
(Sydney Anglo's translation). Duarte adds that roughly horizontal edge blows are more effective in mounted combat than descending vertical blows. He does not say why, but modern physics would say that these blows are better aligned with the vector of the moving horse, and so do a better job of increasing the velocity of the sword relative to the point of impact.
Interestingly, Duarte does not recommend a purely horizontal blow, but a somewhat oblique blow from high to low. Contemporary masters of martial arts understood that downward oblique blows were a natural way for the human body to deliver powerful blows. Germans called them zornhau, the instinctive blow of an angry man. Italians spoke of posta di donna, a position from which even a lady could hit hard.
Duarte's advice represented a wise compromise between the angle that would best align with the movement of the horse and the angle best suited to human biomechanics.
Modeling the physics of a swung sword is complicated. We can grossly oversimplify and say that human muscles can move the percussion point of a sword at about 48 mph. Pretend that the sword is a simple projectile, although the truth is more complicated. Now add that sword and target are galloping towards each other at a net velocity of 27 mph. The kinetic energy of the encounter more than doubles.
When the the Bastard of Burgundy fought Lord Scales in 1467, the Bastard split Scales' visor with an edge blow, making a substantial opening in the metal, three fingers long and wide enough for a grain of wheat to pass through: "en la visiere a cost sy auant fendu que de trois doigs de large et vng gran de bled pouioit passer par la fente dont lespee fut escardee en deux lieux"
Although the horse's momentum could add considerable force to an edge blow, Giovanni Battista Gaiana, writing in 1619, recommended aiming at the opponent's pauldron or vambrace. These pieces were typically thinner than the helmet.