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Kinetic energy (energy of motion)

Kinetic energy is the energy an object possesses due to motion. This article explains its classical formulas, forms, history, applications, and important distinctions including frame dependence and high-speed limits.

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Overview

Kinetic energy is the portion of a system's energy associated with motion. A clear definition frames it as energy an object has because it moves; that motion can be translational, rotational, vibrational, or random at the microscopic scale. Describing how motion relates to energy helps predict the effects of forces and collisions and links to other energy types such as potential energy through conversion and the work-energy principle. The concept applies across mechanics, thermodynamics, and engineering.

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Forms and formulas

In classical mechanics the translational kinetic energy of a particle of mass m moving at speed v is commonly written as KE = 1/2 m v^2. For extended rigid bodies there is also rotational kinetic energy, expressed in an analogous form: KE_rot = 1/2 I ω^2, where I is the moment of inertia and ω the angular speed. Kinetic energy is a scalar measured in joules (kg·m^2/s^2). Its change equals the net work done on the object (the work–energy theorem). Unlike some conserved quantities, kinetic energy depends on the observer's frame of reference; different inertial observers can assign different kinetic energies to the same moving object. See also the role of motion in defining kinetic quantities.

History and development

The idea evolved from early debates about "vis viva" in the 17th–18th centuries and later clarified into the modern energy concept. Work by scientists studying heat, mechanical work and conservation principles established that energy could change form but is conserved overall. The work–energy theorem and formulations for rotational motion were incorporated into classical mechanics textbooks and engineering practice.

Applications and examples

  • Transportation: cars and trains carry kinetic energy that must be managed in braking and collisions.
  • Power generation: wind and hydro turbines convert kinetic energy of a fluid into electricity.
  • Ballistics and sports: projectiles and moving athletes transfer kinetic energy on impact.
  • Thermal physics: temperature relates to the average microscopic kinetic energy of particles in gases and solids.

Key distinctions and notable facts

Kinetic energy is distinct from potential energy, which depends on position or configuration. It is frame-dependent, scalar, and additive across non-interacting parts. At speeds approaching the speed of light the simple 1/2 m v^2 expression becomes inaccurate; relativistic mechanics provides the corrected relation and shows kinetic energy grows much faster as speed increases. Practical engineering uses the classical formula in everyday situations and switches to relativistic formulas only for high-energy particles or astrophysical problems.

Kinetic energy in classical mechanics

Mass point

In classical mechanics, the kinetic energy Eof a mass point depends on its mass mand its velocity v:

E_{{\mathrm {kin}}}={\frac {1}{2}}mv^{2}

For example, if a car of mass traveling {\displaystyle m=1000\,\mathrm {kg} }at a speed of {\displaystyle v=100\,\mathrm {km} /\mathrm {h} }, it therefore has a kinetic energy of {\displaystyle E={\frac {1}{2}}\cdot 1000\,\mathrm {kg} \cdot \left(100\,{\frac {\mathrm {km} }{\mathrm {h} }}\right)^{2}\approx {\frac {1}{2}}\cdot 1000\,\mathrm {kg} \cdot \left(27{,}78\,{\frac {\mathrm {m} }{\mathrm {s} }}\right)^{2}=385\,800\,\mathrm {J} }(the joule, \mathrm{J}, is the SI unit of energy).

If the state of motion of the body is not described by its velocity v, but by its momentum p , as is usual in Hamiltonian mechanics, among other things, then the following applies to the kinetic energy (because p = m v):

{\displaystyle E_{\mathrm {kin} }={\frac {p^{2}}{2m}}}

Simple derivation

If a body of mass m is vaccelerated from rest to velocity , the acceleration work must be Wadded for it. If the force is constant, then:

W = Fs,

where s is the distance travelled in the direction of the force. The force imparts uniform acceleration the body. a, according to the basic equation of mechanics, F=ma. After a time , tthe velocity v=at, and the distance has been {\displaystyle s={\tfrac {1}{2}}at^{2}}traveled. Putting everything in above gives the acceleration work

{\displaystyle W=ma\cdot {\frac {1}{2}}at^{2}={\frac {1}{2}}mv^{2}}.

Since the kinetic energy at rest has the value zero, it reaches exactly this value after the acceleration processW. Consequently, for a body of mass mwith velocity v:

{\displaystyle E_{\mathrm {kin} }={\frac {1}{2}}mv^{2}}

Movement in a coordinate system

If one describes the motion of a body in a coordinate system, the kinetic energy can be calculated like this, depending on the choice of the coordinate system:

{\displaystyle E_{\mathrm {kin} }={\frac {1}{2}}m\left({\dot {x}}^{2}+{\dot {y}}^{2}+{\dot {z}}^{2}\right)}

{\displaystyle E_{\mathrm {kin} }={\frac {1}{2}}m\left({\dot {r}}^{2}+r^{2}{\dot {\varphi }}^{2}\right)}

{\displaystyle E_{\mathrm {kin} }={\frac {1}{2}}m\left(r^{2}\left[{\dot {\vartheta }}^{2}+{\dot {\varphi }}^{2}\sin ^{2}\vartheta \right]+{\dot {r}}^{2}\right)}

  • Cylinder coordinates ( r, \varphi, z ):

{\displaystyle E_{\mathrm {kin} }={\frac {1}{2}}m\left({\dot {r}}^{2}+r^{2}{\dot {\varphi }}^{2}+{\dot {z}}^{2}\right)}

Here the point above the coordinate means its change in time, the derivative with respect to time. The formulas do not take into account the energy that may be in the body's own rotation.

Rigid bodies

The kinetic energy of a rigid body with total mass Mand velocity v_\mathrm{s}its center of gravity is the sum of the energy from the motion of the center of gravity (translational energy) and the rotational energy from the rotation about the center of gravity:

{\displaystyle E_{\mathrm {kin} }={\frac {1}{2}}M{v_{\mathrm {s} }}^{2}+{\frac {1}{2}}J_{\mathrm {s} }\omega ^{2}}

Here J_\mathrm{s}the moment of inertia of the body with respect to its center of gravity and ω is \omega the angular velocity of the rotation.

With the inertia tensor I, this is generally written as:

{\displaystyle E_{\mathrm {kin} }={\frac {1}{2}}M{v_{\mathrm {s} }}^{2}+{\frac {1}{2}}{\boldsymbol {\omega }}^{T}I{\boldsymbol {\omega }}}

Hydrodynamics

In hydrodynamics, the kinetic energy density is often given instead of the kinetic energy. This is usually expressed by a small eor : \epsilon

{\displaystyle e_{\mathrm {kin} }={\frac {E_{\mathrm {kin} }}{V}}={\frac {1}{2}}\rho v^{2}}

Here ρ \rho denotes the density and V the volume.

Kinetic energy in relativistic mechanics

In relativistic physics, the above dependence of kinetic energy on velocity applies only approximately for velocities much smaller than the speed of light. From the approach that the kinetic energy E_{\mathrm {kin} }the difference of total energy and rest energy follows:

{\displaystyle E_{\mathrm {kin} }=\gamma mc^{2}-mc^{2}=\left(\gamma -1\right)mc^{2}}

Where cthe speed of light, mmass, and γ \gamma is the Lorentz factor.

{\displaystyle \gamma ={\frac {1}{\sqrt {1-(v/c)^{2}}}}.}

From the Taylor expansion to v/cobtains

{\displaystyle E_{\mathrm {kin} }={\frac {1}{2}}mv^{2}+{\frac {3}{8}}{\frac {mv^{4}}{c^{2}}}+\cdots },

so for v\ll cagain the Newtonian kinetic energy.

Since the energy would have to grow beyond all limits as the velocity goes toward the speed of light, {\displaystyle \lim _{v\to c}E_{\mathrm {kin} }=\infty ,}it is not possible to accelerate a massed body to the speed of light.

The diagram on the right shows, for a body with mass m = 1\, \mathrm{kg}, the relativistic and Newtonian kinetic energy as a function of velocity (measured in multiples of the speed of light).

Since the velocity of a moving body depends on the reference frame, this also applies to its kinetic energy. This is true in Newtonian and relativistic physics.

Application examples

→ Main article: Tests of the relativistic energy-momentum relation

In the electric field, the energy of an electron of charge eand mass increases mlinearly with the accelerating voltage U passed through. The kinetic energy is now the difference of the total relativistic energy Eand the rest energy E0. The kinetic energy eUis thus:

e\cdot U=E-E_{0}

If one takes into account that for the total energy

E^{2}=c^{2}p^{2}+E_{0}^{2}\quad (*)

holds ( p: relativistic momentum) and between momentum and total energy the relation

cp=E\cdot {\frac {v}{c}}

consists, it follows for the total energy from (*)thus:

E(v)={\frac {E_{0}}{{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}}

Calculating the difference between E(v)and E_{0}, set the expression equal to e\cdot Uand resolve to , we vfinally obtain:

v=c\cdot {\sqrt {1-{\left({\frac {1}{1+{\frac {eU}{E_{0}}}}}\right)}^{2}}}with the rest energy of an electron E_{0}=0{,}51\,{\mathrm {MeV}}

For acceleration voltages below 1 kV, the velocity can be estimated from the classical approach for the kinetic energy; at higher energies, relativistic calculations must be made. Already at a voltage of 10 kV, the electrons reach a velocity of almost 20 % of the speed of light, at 1 MV 94 %.

The Large Hadron Collider supplies protons with a kinetic energy of 6.5 TeV. This energy is about 8 thousand times greater than the rest energy of a proton. A collision between oppositely accelerated protons can produce particles with a correspondingly high rest energy.

Comparison of relativistic and classical kinetic energy
Comparison of relativistic and classical kinetic energy
Relativistic velocity of an electron after passing through an electric field
Relativistic velocity of an electron after passing through an electric field

Questions and answers

Q: What is kinetic energy?

A: Kinetic energy is the energy possessed by an object as a result of its motion.

Q: Can kinetic energy be converted into other forms of energy?

A: Yes, kinetic energy can be converted into other forms of energy such as gravitational or electric potential energy.

Q: What is gravitational potential energy?

A: Gravitational potential energy is the energy that an object has because of its position in a gravitational field.

Q: What is electric potential energy?

A: Electric potential energy is the energy that an object has because of its position in an electric field.

Q: How is kinetic energy related to motion?

A: Kinetic energy is related to motion because its presence arises from the fact that an object is moving.

Q: Can objects possess kinetic energy when they are at rest?

A: No, objects cannot have kinetic energy when they are at rest. Kinetic energy is only present when objects are in motion.

Q: What happens to the kinetic energy of an object when it is converted into other forms of energy?

A: When kinetic energy is converted into other forms of energy, the amount of kinetic energy decreases while the amount of the new form of energy increases.

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