The Measure of Motion – Work¶
Core Argument¶
Chapter Digest: The Measure of Motion – Work¶
This chapter dissects a long-standing controversy in classical mechanics: whether the true measure of motion is mv (momentum), as Descartes argued, or mv² (vis viva), as Leibniz maintained. The author traces Leibniz’s derivation of mv² from the law of falling bodies, noting that a body falling four feet attains twice the velocity of one falling one foot, yet requires four times the force to be raised again. Leibniz further demonstrated that adopting mv would logically imply the possibility of a perpetual motion machine. D’Alembert’s attempted reconciliation—assigning mv to equilibrium and mv² to retarded motion—is dismissed as a mathematical blunder. Later mechanicians accepted both measures without resolving the underlying contradiction, applying mv to statics and mv² to dynamics.
The chapter argues that the two measures are not contradictory but apply to different aspects of motion. The measure mv applies to transferred, lasting mechanical motion, as in levers; mv² measures mechanical motion that vanishes as such—that is, motion that is converted into another form. On impact of inelastic bodies, mv fails because it cannot account for the motion that is lost, whereas mv² correctly expresses the quantity of motion that disappears. This vanishing motion is not annihilated but transformed into potential energy, heat, or electricity, in proportion to mv²/2. A 24-lb cannon-ball striking armour demonstrates this: its mechanical motion disappears and reappears as heat, raising the shot to red heat. Friction and magneto-electrical machines provide further evidence.
Thus, mv measures mechanical motion as mechanical motion, while mv²/2 measures it by its capacity to become another form. The concept of work is defined as the change of form of motion regarded quantitatively. Helmholtz is criticised for failing to grasp this, treating friction and inelastic impact as destroying work when in fact mechanical work is performed and motion converted. He adopted mv²/2 as the measure of vis viva merely for convenience, unaware that he had transformed a proportional measure into an absolute one. The connection between vis viva and work was accepted only for simplicity of calculation, not from genuine comprehension.