Proving the first law of thermodynamics from energy conservation

Step 1 of 4: Starting point: work converts to kinetic energy

Intuition

Push a rolling cart: the force of your hands acting over a distance converts into extra speed of the cart — work is the 'money' crossing the mechanical boundary.

A = \Delta K \quad (\text{định lý động năng})

Analysis

For a particle, the work–energy theorem — a direct corollary of Newton's second law: $A = \int \vec{F}\cdot d\vec{s} = \int m\,dv = \tfrac12 m v_2^2 - \tfrac12 m v_1^2 = \Delta K$ . The work done on the body equals exactly its kinetic-energy gain.