Émilie du Châtelet · Science
Du Châtelet’s championing of <em>vis viva</em> (living force, mass times velocity squared) against the Newtonians’ momentum (mass times velocity): the 's Gravesande clay-ball experiments, her argument in the <em>Institutions</em>, and the long road from mv² to kinetic energy and the square in <em>E = mc²</em>.
In the first half of the eighteenth century, one of the fiercest disputes in physics sounded almost like a quarrel over bookkeeping: how do you measure the ‘force’ of a moving body? Descartes, and after him the Newtonians, held that the relevant quantity is mass times velocity (mv) - what we now call momentum. Leibniz insisted that the true measure of a body’s force is mass times the square of its velocity (mv²), which he named vis viva, the ‘living force.’ The difference looks tiny and technical. It is not. Hidden inside it is the birth of the concept of energy. A body moving twice as fast has twice the momentum but four times the living force; three times as fast, three times the momentum but nine times the living force. Which number captures what a moving body can actually do? Du Châtelet devoted years to arguing that the answer is the square.
The most persuasive evidence came from the Dutch Newtonian Willem 's Gravesande, who dropped brass balls into a bed of soft clay from different heights and measured how deep they sank. A ball dropped from four times the height arrives (by Galileo’s law of fall) with twice the velocity - and it sank not twice but four times as deep. From nine times the height, three times the velocity, nine times the depth. The depth of the impression - a fair measure of the work the ball did flattening the clay - tracked the square of the velocity exactly. Strikingly, 's Gravesande had begun as a convinced Newtonian expecting to confirm mv, and his own experiments converted him to vis viva. Du Châtelet seized on these results and reported them in her Institutions de physique as experimental proof that the living force, mv², is the quantity that measures a body’s power to act.
Du Châtelet did not have the modern concept of energy; nobody did. But the quantity she fought for, mv², is its direct ancestor. Multiply the living force by one half and you have the modern kinetic energy, ½mv² - the energy a body has by virtue of its motion, the very thing 's Gravesande’s clay was measuring. The insistence that the square of the velocity, not velocity itself, is the physically fundamental quantity was the crucial insight, and it never went away. The square of the speed sits at the heart of kinetic energy; and the square of a speed - the speed of light - sits at the heart of the most famous equation in physics, Einstein’s E = mc². When du Châtelet defended mv² against the momentum party, she was defending, without quite knowing it, the squared term that would run straight through the physics of the next two centuries.
This is the opening of the lesson. The rest — the dialogue, the primary source, and the recall — is in the app.
You learned that du Châtelet defended Leibniz’s vis viva (mv²) against the Cartesian and Newtonian measure of force (mv), using 's Gravesande’s clay-ball experiments, and that mv² is the ancestor of kinetic energy. Explain the experiment and why the depth of the dent grows as the square of the speed.
Leads to Gottfried Leibniz.
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