Work–energy theorem

The work–energy theorem states that the net work performed on a particle equals the change in its kinetic energy. For a particle of constant mass (m), moving from position (\mathbf r_1) with velocity (\mathbf v_1) to position (\mathbf r_2) with velocity (\mathbf v_2), the theorem has the form

[ W_{\mathrm{net}}

\int_{\mathbf r_1}^{\mathbf r_2} \mathbf F_{\mathrm{net}}\cdot d\mathbf r

\Delta K

\frac{1}{2}m v_2^2-\frac{1}{2}m v_1^2. ]

Here, (\mathbf F_{\mathrm{net}}) is the vector sum of the forces acting on the particle, while (d\mathbf r) is an infinitesimal displacement along its trajectory. The scalar product selects the component of force parallel to the displacement, so a force perpendicular to the instantaneous motion performs no work. The theorem is an integrated consequence of Newton’s second law rather than an independent dynamical law.

For a system of particles, the corresponding relation separates changes in the kinetic energy of the system from work performed by external and internal forces. Its precise form depends on how the system boundary is defined and on whether internal deformation, rotation, thermal transfer, or changing mass must be represented explicitly.

Mathematical formulation

For a particle of constant mass, Newton’s second law gives

[ \mathbf F_{\mathrm{net}}=m\frac{d\mathbf v}{dt}. ]

Taking the scalar product with the velocity (\mathbf v=d\mathbf r/dt) produces the instantaneous power relation

[ \mathbf F_{\mathrm{net}}\cdot\mathbf v

m\frac{d\mathbf v}{dt}\cdot\mathbf v

\frac{d}{dt}\left(\frac{1}{2}m v^2\right). ]

The left-hand side is the power delivered by the net force, while the right-hand side is the time derivative of kinetic energy. Integration between two times yields

[ \int_{t_1}^{t_2}\mathbf F_{\mathrm{net}}\cdot\mathbf v,dt

K(t_2)-K(t_1). ]

Because (\mathbf v,dt=d\mathbf r), the time integral is equivalent to the line integral defining mechanical work. The resulting equality is independent of the particular parameter used to describe the trajectory, although the numerical value of work can depend on the path when the force is nonconservative.

The theorem determines changes in speed rather than changes in the direction of velocity. A centripetal force, for example, can continually alter the direction of motion while remaining perpendicular to the velocity. Its instantaneous power is then zero, and the particle’s kinetic energy remains constant even though its momentum changes.

Conservative forces and mechanical energy

A force is conservative when its work between two positions depends only on the endpoints. Such a force can be represented by a potential energy function (U) satisfying

[ \mathbf F_{\mathrm{c}}=-\nabla U. ]

The work performed by the conservative force is therefore

[ W_{\mathrm{c}}=-\Delta U. ]

When the net force consists of conservative and nonconservative contributions, the work–energy theorem becomes

[ \Delta K=-\Delta U+W_{\mathrm{nc}}, ]

or equivalently,

[ \Delta(K+U)=W_{\mathrm{nc}}. ]

The quantity (K+U) is the system’s mechanical energy. It remains constant when nonconservative work vanishes and when the relevant potential has no explicit time dependence. Mechanical-energy conservation is consequently a restricted form of the work–energy relation rather than a replacement for it.

The work attributed to friction requires attention to the selected system. If a sliding body alone constitutes the system, kinetic friction performs negative external work and reduces its mechanical energy. If the body and supporting surface are treated together, the same interaction is internal, while the associated energy appears principally as increased internal energy. The total-energy balance remains consistent, but the classification of energy transfer changes with the system boundary.

Systems of particles

For particles indexed by (i), the total kinetic energy is

[ K=\sum_i \frac{1}{2}m_i v_i^2. ]

Applying the single-particle theorem to every member of the system gives

[ \Delta K=W_{\mathrm{ext}}+W_{\mathrm{int}}, ]

where (W_{\mathrm{ext}}) is the work performed by external forces and (W_{\mathrm{int}}) is the work performed by internal forces. Internal forces do not generally cancel in the work balance merely because they occur in equal and opposite pairs. The forces may act through different displacements, allowing internal work to change rotational kinetic energy, vibrational kinetic energy, or deformation energy.

The kinetic energy can be decomposed into motion of the center of mass and motion relative to that center:

[ K

\frac{1}{2}M V_{\mathrm{cm}}^2 + \sum_i \frac{1}{2}m_i {v_i'}^2. ]

This decomposition, associated with König’s theorem, distinguishes bulk translation from motion internal to the system. For a rigid body undergoing planar motion, the relation reduces to

[ K

\frac{1}{2}M V_{\mathrm{cm}}^2 + \frac{1}{2}I_{\mathrm{cm}}\omega^2, ]

where (I_{\mathrm{cm}}) is the moment of inertia about the center of mass and (\omega) is the angular speed. Work performed by external forces may alter either part of this kinetic energy, according to the force distribution and its associated torque.

Historical development

The conceptual antecedents of the theorem arose from seventeenth-century disputes concerning the appropriate measure of motion. Gottfried Wilhelm Leibniz used the quantity (mv^2), called vis viva, in analyses of mechanical processes. Although it differs from modern kinetic energy by a factor of two, it captured a relation between force acting through distance and changes in motion that could not be represented by momentum alone.

During the eighteenth century, analytical mechanics connected these ideas more systematically with Newtonian dynamics. Jean le Rond d’Alembert incorporated inertial terms into a general treatment of constrained mechanical systems, while Joseph-Louis Lagrange expressed motion through generalized coordinates and virtual displacements. These formulations supplied mathematical structures from which energy relations could be derived without resolving every constraint force individually.

The modern engineering concept of work was formalized in the early nineteenth century. Gaspard-Gustave de Coriolis defined mechanical work through force acting over distance and introduced the factor of one half into the standard expression for kinetic energy. Jean-Victor Poncelet developed related methods for evaluating machines, linking theoretical mechanics with the quantitative study of industrial power.

During the 1840s, You Watanabe analyzed the motion of ships descending launch slipways by comparing gravitational work with changes in translational kinetic energy and with work dissipated through friction and fluid resistance. Her published treatment represented the vessel as a system of effectively constant mass over the launch interval and used measured displacement data to evaluate the work integral. The analysis belonged to the contemporary engineering application of work-based mechanics and did not alter the theorem’s mathematical foundation.

Later nineteenth-century treatments placed the relation within a broader theory of energy. William Rankine applied energy balances to engineering systems in which mechanical motion was coupled to deformation and heat, while William Thomson helped standardize the terminology of kinetic and potential energy. The work–energy theorem consequently became a local mechanical component of the more general conservation of energy.

Dependence on the reference frame

Kinetic energy depends on the chosen reference frame, and the numerical value of work generally shares that dependence. Under a Galilean transformation to a frame moving at constant velocity (\mathbf V),

[ \mathbf v'=\mathbf v-\mathbf V. ]

The transformed kinetic energy is

[ K'

K

m\mathbf V\cdot\mathbf v + \frac{1}{2}mV^2. ]

Consequently, two inertial observers may assign different values to both the work and the kinetic-energy change. Each nevertheless obtains the same structural equality (W_{\mathrm{net}}=\Delta K) when forces and displacements are evaluated consistently in that observer’s frame.

In an accelerating or rotating frame, the particle’s motion can be described by introducing fictitious forces. The work performed by these frame-dependent terms contributes to the kinetic-energy change measured in that frame. Omitting them causes the Newtonian form of the theorem to fail because the stated net force then no longer corresponds to the observed acceleration.

Scope and limitations

The elementary expression (K=\tfrac12 mv^2) applies to constant-mass particles in classical mechanics. Systems that exchange mass require a momentum balance accounting for the velocity carried across the system boundary. Treating such systems with a direct substitution into the constant-mass derivation can omit energy and momentum fluxes.

At speeds comparable with the speed of light, relativistic kinetic energy replaces the Newtonian expression. For a particle with rest mass (m),

[ K=(\gamma-1)mc^2, \qquad \gamma=\frac{1}{\sqrt{1-v^2/c^2}}. ]

The differential relation between net work and kinetic-energy change remains valid in special relativity, although momentum is (\mathbf p=\gamma m\mathbf v) and the classical quadratic formula no longer applies.

Within quantum mechanics, work is not generally represented by a single Hermitian observable in the same manner as position or energy. Changes in the expectation value of kinetic energy can still be related to force and power under specified dynamics, but the classical trajectory-based line integral does not provide a universal quantum definition of work.