• Émilie du Châtelet Vis Viva pixel-art physics sticker, 3" × 3" / White

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Émilie du Châtelet Sticker

2× The Speed. 4× The Dent. | Physics

\[\begin{gathered} E_k = \tfrac{1}{2} m v^{2} \\ (\text{vis viva: } m v^{2}) \end{gathered}\]
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Twice the speed, four times the dent. In Institutions de physique (1740), Émilie du Châtelet championed vis viva: a moving body’s impact grows with mass × speed². A pixel-art portrait on a 3" × 3" sticker; the physics panel below has the equation, every symbol and the sources.

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Size3″ × 3″

3″ × 3″ kiss-cut sticker.

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The physics

Kinetic energy grows with the square of speed, so twice the speed carries four times the energy. Du Châtelet used experiments with balls dropped into clay, where the dent depth grew with the square of the speed, to argue for this 'vis viva' measure.

The sticker says 2× The Speed. 4× The Dent. In the 1720s that was an argument conducted in clay.

Natural philosophers of the early 1700s fought over what a moving body carries. Followers of Descartes measured it as mass times speed. Followers of Leibniz said mass times speed squared, which Leibniz called vis viva, living force.

In Leiden, Willem 's Gravesande put it to the test by dropping balls into soft clay, after similar work by Giovanni Poleni. Balls of the same size but different masses made identical dents when their drop heights were inversely proportional to their masses. The height needed grows as speed squared, so equal dents meant equal mass times speed squared, not equal mass times speed. A ball arriving twice as fast does the same damage as one four times as heavy.

Émilie du Châtelet put this experiment at the heart of the chapter on the force of bodies in her Institutions de physique (1740). She describes a ball of mass 3 at speed 1 beside a ball of mass 1 at speed 3, the faster making by far the larger dent, and argues for vis viva. She reported and championed the experiments; they were 's Gravesande's. On this sticker she drops the balls herself, which is our dramatic licence.

Clay did not really settle the quarrel. Momentum and energy are both conserved, both useful, and measure different things. Mass times speed squared survives, halved, as kinetic energy. Du Châtelet also translated Newton's Principia into French. It was published in 1759, ten years after her death, and is still the leading French translation.

The dents here are drawn at exactly one to four. Double the speed, quadruple the dent.

Equations

\[\begin{gathered} E_k = \tfrac{1}{2} m v^{2} \\ (\text{vis viva: } m v^{2}) \end{gathered}\]
\[v = \sqrt{2 g h}\;\Rightarrow\; h \propto v^{2}\]
\[\begin{gathered} m g h \approx F d\;\Rightarrow\; d \propto v^{2} \\ (\text{idealised: constant} \\ \text{clay resistance}) \end{gathered}\]
\[\begin{gathered} m_1 h_1 = m_2 h_2 \\ \Leftrightarrow\; m_1 v_1^{2} = m_2 v_2^{2} \\ \Rightarrow\; \text{equal dents} \\ (\text{'s Gravesande}) \end{gathered}\]
SymbolMeaningUnit
\(E_k\) kinetic energy \(\mathrm{J}\)
\(m\) mass \(\mathrm{kg}\)
\(v\) impact speed \(\mathrm{m/s}\)
\(g\) gravitational acceleration \(\mathrm{m/s²}\)
\(h\) drop height \(\mathrm{m}\)
\(F\) average resisting force of the clay (assumed constant) \(\mathrm{N}\)
\(d\) dent depth \(\mathrm{m}\)
\(m_1, m_2\) masses of two balls of the same size \(\mathrm{kg}\)
\(h_1, h_2\) their drop heights \(\mathrm{m}\)
\(v_1, v_2\) their impact speeds \(\mathrm{m/s}\)

Sources

  1. Du Châtelet, Institutions de physique (1740), ch. 21 'Of the Force of Bodies', §584 — English translation, History of Women Philosophers and Scientists
  2. Wikipedia — Willem 's Gravesande (clay experiments; Poleni)
  3. Stanford Encyclopedia of Philosophy — Émilie du Châtelet
  4. Wikipedia — Vis viva

Source Du Châtelet, G.-É. (1740). Institutions de physique, Ch. 21 (De la force des corps). Paris: Prault.

Field notes

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