Kinetic energy is the energy a body or particle possesses because of its motion relative to a specified observer. In classical mechanics, a particle of mass (m) moving at speed (v) has kinetic energy (K=\tfrac12mv^2). This quantity connects motion with the work performed by forces. It can describe the translation of a vehicle, the rotation of a wheel, or the microscopic motion of gas particles. Kinetic energy depends on the observer’s reference frame rather than being an intrinsic, frame-independent property of an object. (openstax.org)
Classical definition and measurement
For a nonrelativistic particle with constant mass,
[ K=\frac12mv^2=\frac{\mathbf p\cdot\mathbf p}{2m}, ]
where (\mathbf p=m\mathbf v) is its momentum. Kinetic energy is a scalar: it has magnitude but no spatial direction. It is nonnegative and becomes zero when the particle is stationary in the chosen frame. Objects moving in opposite directions can therefore have identical kinetic energies, although their momenta differ. (openstax.org)
In the International System of Units, kinetic energy is measured in joules, with (1\ \mathrm J=1\ \mathrm{kg,m^2,s^{-2}}). At fixed speed, doubling mass doubles kinetic energy; at fixed mass, doubling speed quadruples it. For example, a (2)-kilogram object moving at (3) metres per second has (9) joules of kinetic energy. These scaling relationships follow directly from the classical formula. (openstax.org)
Work and changes in motion
The work–energy theorem states that the net work performed on a particle equals its change in kinetic energy:
[ W_{\mathrm{net}}=\Delta K=K_{\mathrm f}-K_{\mathrm i}. ]
For a force that varies along a trajectory, work is the line integral
[ W_{\mathrm{net}}=\int \mathbf F_{\mathrm{net}}\cdot d\mathbf r. ]
Using Newton’s second law for constant mass gives (dW=m\mathbf v\cdot d\mathbf v); integration yields the factor of one-half in the kinetic-energy expression. Positive net work increases speed, while negative net work decreases it. A force perpendicular to the instantaneous velocity changes the direction of motion without changing kinetic energy. (openstax.org)
This theorem often determines a final speed without requiring a complete calculation of the trajectory or the time-dependent acceleration. All forces acting on the particle must be included when calculating net work. For extended, deformable systems, energy accounting must additionally distinguish bulk motion from rotation and internal changes. (openstax.org)
Reference frames and rotation
Because velocity depends on the reference frame, observers moving relative to one another generally assign different kinetic energies to the same object. A passenger stationary inside a uniformly moving train has zero translational kinetic energy relative to the train but nonzero kinetic energy relative to the ground. Changing frames changes the measured motion, not the physical object. (openstax.org)
An extended body may possess both translational and rotational kinetic energy. For a rigid body rotating about a fixed axis,
[ K_{\mathrm{rot}}=\frac12I\omega^2, ]
where (I) is its moment of inertia about that axis and (\omega) is its angular speed. Unlike mass alone, moment of inertia reflects how mass is distributed relative to the axis. (openstax.org)
For a body rolling without slipping, its total kinetic energy can be written
[ K=\frac12MV_{\mathrm{cm}}^2+\frac12I_{\mathrm{cm}}\omega^2. ]
The two terms describe translation of the centre of mass and rotation around it. A flywheel stores energy in the rotational term, illustrating how kinetic energy can serve as a mechanically recoverable energy store. (openstax.org)
Energy conversion and collisions
Kinetic energy is not generally conserved by itself. Under suitable conditions, it exchanges with potential energy while their sum, mechanical energy, remains constant. A falling object, for example, gains kinetic energy as gravitational potential energy decreases. Friction can instead transfer mechanical energy into microscopic forms, increasing internal energy. Such conversion is compatible with conservation of total energy. (openstax.org)
An elastic collision conserves the total kinetic energy of the colliding system between its initial and final states. An inelastic collision does not, although total momentum remains conserved when external impulse is negligible. Objects that stick together undergo a perfectly inelastic collision; part of their initial kinetic energy can become internal energy or deformation energy. Simply bouncing apart does not establish that a collision is elastic. (openstax.org)
Microscopic motion and temperature
In statistical mechanics, microscopic kinetic energy connects particle motion with macroscopic temperature. For a classical ideal gas in thermal equilibrium, the average translational kinetic energy per molecule is
[ \langle K_{\mathrm{trans}}\rangle=\frac32k_{\mathrm B}T, ]
where (k_{\mathrm B}) is the Boltzmann constant and (T) is absolute temperature. At equal temperatures, particles of different masses have equal average translational kinetic energies, but different characteristic speeds. (openstax.org)
Molecules can also possess rotational and vibrational energy. Consequently, the internal energy of matter is not generally identical to its translational kinetic energy, and temperature is not a universal measure of every form of microscopic motion. Quantum effects can limit the excitation of rotational or vibrational modes. (openstax.org)
Relativistic and quantum descriptions
In the special theory of relativity, a particle with nonzero rest mass has
[ K=(\gamma-1)mc^2,\qquad \gamma=\frac{1}{\sqrt{1-v^2/c^2}}, ]
where (c) is the speed of light. Its total energy is (\gamma mc^2); subtracting the rest energy (mc^2), associated with mass–energy equivalence, gives kinetic energy. At speeds much smaller than (c), this reduces to (\tfrac12mv^2). As (v) approaches (c), the required kinetic energy grows without bound. (openstax.org)
In nonrelativistic quantum mechanics, kinetic energy is represented by an operator. For a particle of constant mass without an electromagnetic vector potential,
[ \hat K=\frac{\hat{\mathbf p}^{,2}}{2m} =-\frac{\hbar^2}{2m}\nabla^2. ]
Here (\hbar) is the reduced Planck constant. Acting on a wave function, this operator supplies the kinetic term in the Schrödinger equation. Its mathematical form retains the classical momentum-squared relationship while replacing momentum with a quantum operator. (ocw.mit.edu)