= Solution
The strong evidence is the reproducible conversion between mechanical <work> and thermal changes. In <Joule's paddle-wheel experiment>, descending masses drive vanes in a liquid, and a measured input of mechanical <energy> produces a measured rise in <temperature>. Allowing for the vessel and thermal losses gives a quantitative mechanical equivalent. Sustained frictional heating, including <Rumford's cannon-boring experiment>, also challenged a simple account in which rubbing merely released a fixed stored supply of a material heat fluid.<On the Mechanical Equivalent of Heat by James Prescott Joule>[1850, friction experiments and calorimetric corrections]{ref}
Electrical and compressional experiments reinforce the connection. A resistor warms under <Joule heating>, with electrical input $\int VI\,dt$. A gas compressed with little <heat transfer> can rise in <temperature>, showing that mechanical <work> can increase its <internal energy>. Conversely an expanding gas can perform <work> while cooling. Such experiments connect several kinds of change through a conserved <energy> balance.
The <kinetic theory of gases> supplies a microscopic model. Translational molecular motion produces momentum transfer to walls; for a dilute monatomic ideal gas,
$$
p=\frac13\frac NV m\langle v^2\rangle,
\qquad U=\frac32Nk_{\rm B}T.
$$
Thus increasing molecular kinetic <energy> is reflected in <temperature> and pressure changes. These formulas describe a specified model, not a proof that every substance's <internal energy> consists only of translational motion. Rotations, vibrations and intermolecular potential <energy> also matter.
In modern terminology, \b[<heat> is energy transferred because of a temperature difference]. The body has <internal energy>, and microscopic motion accounts for part of it. Calling heat “a form of motion” captures the historical rejection of a conserved material caloric, but should not obscure that distinction between stored <energy> and an energy-transfer process.
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