Mass Kinetic Energy from Virtual Work

From Newton's second law,
(introduced in Eq.
(B.1)),
we can use d'Alembert's idea of *virtual work* to derive the
formula for the kinetic energy of a mass given its speed
.
Let
denote a small (infinitesimal) displacement of the mass in
the
direction. Then we have, using the calculus of differentials,

Thus, by Newton's second law, a differential of work applied to a mass by force through distance boosts the kinetic energy of the mass by . The kinetic energy of a mass moving at speed is then given by the integral of all such differential boosts from 0 to :

where denotes the

The quantity
is classically called the *virtual work*
associated with force
, and
a *virtual displacement*
[546].

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