### 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 :

*kinetic energy*of mass traveling at speed .

The quantity is classically called the *virtual work*
associated with force , and a *virtual displacement*
[544].

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Energy in the Mass-Spring Oscillator

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Kinetic Energy of a Mass