# Kinetic energy

The kinetic energy (also called in the old writings screw viva , or lifeblood ) is the energy which a body because of its movement has. The kinetic energy of a body is equal to the work necessary to make pass says it body of the rest to its rotation or translatory movement.

It is Guillaume d' Ockham (1280 - 1349) which introduced, in 1323, the difference between what is called the dynamic movement (that we generate) and the kinetic movement (generated by interactions, of which collisions).

## Definitions

### Case of a Not material

In the field of validity of the Newtonian Mechanical , the concept of energy kinetic can be easily highlighted, for a body considered as specific (or Point material ) of constant mass m .

Indeed the fundamental relation of dynamics écrit :

$m \ frac \left\{\ vec \left\{FD\right\}\right\} \left\{dt\right\} = \ sum \ vec \left\{F\right\}$, with $\ sum \ vec \left\{F\right\}$ nap of the forces applied to the material point of mass m (including the " Inertias " in the case of a reference frame not galiléen).

By taking to the scalar product member with member by speed $\ vec \left\{v\right\}$ of the body, it comes:

$m \ left \left(\ frac \left\{\ vec \left\{FD\right\}\right\} \left\{dt\right\} \ right\right) \ cdot \ vec \left\{v\right\} = \ left \left(\ sum \ vec \left\{F\right\} \ right\right) \ cdot \ vec \left\{v\right\}$ , but $\ left \left(\ frac \left\{\ vec \left\{FD\right\}\right\} \left\{dt\right\} \ right\right) \ cdot \ vec \left\{v\right\} = \ frac \left\{D\right\} \left\{dt\right\} \ left \left(\ frac \left\{1\right\} \left\{2\right\} v^ \left\{2\right\} \ right\right)$, it comes as follows: $\ frac \left\{D\right\} \left\{dt\right\} \ left \left(\ frac \left\{1\right\} \left\{2\right\} mv^ \left\{2\right\} \ right\right) = \ sum \ left \left(\ vec \left\{F\right\} \ cdot \ vec \left\{v\right\} \ right\right)$.

One highlights in the member of left the quantity $E_ \left\{K\right\} \ equiv \ frac \left\{1\right\} \left\{2\right\} mv^ \left\{2\right\}$ called kinetic energy of the material point, whose variation is equal to the sum of the power s $\ vec \left\{F\right\} \ cdot \ vec \left\{v\right\}$ of the forces applied to the body ( theorem of the kinetic energy , form " instantanée").

One can obtain a more general expression while considering than one thus has $\ int D \ left \left(\ frac \left\{1\right\} \left\{2\right\} mv^ \left\{2\right\} \ right\right) = \ int m \ vec \left\{v\right\} \ cdot \ vec \left\{FD\right\}$, since $d \left(v^ \left\{2\right\}\right) =2 \ vec \left\{v\right\} \ cdot \ vec \left\{FD\right\}$. By introducing the infinitesimal variation of the Momentum of the body, $\ vec \left\{dp\right\} \ equiv m \ vec \left\{FD\right\}$, it comes to final the expression: $E_k = \ int \ vec \left\{v\right\} \ cdot \ vec \left\{dp\right\}$.

### Case of a system of points

In the case of a body which one cannot consider specific, it is possible to compare it to a system (of an infinity) of material points $M_i$ of masses $m_i$ with $M= \ sum_ \left\{I\right\} m_ \left\{I\right\} \ qquad$ total mass of the body.

The kinetic energy $E_ \left\{K\right\}$ of the system of points can be then simply defined as the sum of the kinetic energies associated with the material points constituting the system: $E_ \left\{K\right\} = \ sum_ \left\{I\right\} E_ \left\{K, I\right\} = \ sum_ \left\{I\right\} \ frac \left\{1\right\} \left\{2\right\} m_ \left\{I\right\} v_ \left\{I\right\} ^ \left\{2\right\}$, (1). This expression is general and does not prejudge nature of the system, deformable or not.

Note:: by considering the limit of the continuous mediums one has $E_ \left\{K\right\} = \ int_ \left\{\left(S\right)\right\} \ frac \left\{1\right\} \left\{2\right\} \ rho \ \left(M\right) v_ \left\{M\right\} ^ \left\{2\right\} D \ tau \$, M being a point running of the system (S).

## Unit

The legal unit is the Joule. Calculations are carried out with the masses in kg and speeds in meters a second.

## Theorem of König

The expression (1) is hardly usable directly, although general. It is possible to rewrite it in another form, whose physical interpretation is easier.

### Statement

reference frame barycentric (or of the Center of mass), noted (R*) , associated with (R). This last is defined like the reference frame in translation compared to (R), and such as the Quantité of movement $\ vec \left\{P^ \left\{*\right\}\right\}$ of the system in (R*) is null. speeds enter the S (R) and (R*) in translation, one a: $\ vec \left\{v_ \left\{I\right\}\right\} = \ vec \left\{v_ \left\{I\right\} ^ \left\{*\right\}\right\} + \ vec \left\{v_ \left\{G\right\}\right\}$, with G center of mass of (S). While susbstituant in (1) it comes:

$E_ \left\{K\right\} = \ frac \left\{1\right\} \left\{2\right\} \ sum_ \left\{I\right\} m_ \left\{I\right\} \ left \left(\ vec \left\{v_ \left\{I\right\} ^ \left\{*\right\}\right\} + \ vec \left\{v_ \left\{G\right\}\right\} \ right\right) ^ \left\{2\right\} = \ frac \left\{1\right\} \left\{2\right\} \ sum_ \left\{I\right\} m_ \left\{I\right\} v_ \left\{I\right\} ^ \left\{*2\right\} + \ left \left(\ sum_ \left\{I\right\} m_ \left\{I\right\} \ vec \left\{v_ \left\{I\right\} ^ \left\{*\right\}\right\} \ right\right) \ cdot \ vec \left\{v_ \left\{G\right\}\right\} + \ frac\left\{1\right\} \left\{2\right\} \ left \left(\ sum_ \left\{I\right\} m_ \left\{I\right\} \ right\right) v_ \left\{G\right\} ^ \left\{2\right\}$,

however there is $M= \ sum_ \left\{I\right\} m_ \left\{I\right\} \ qquad$ total mass of the body and by definition of (R*), $\ vec \left\{P^ \left\{*\right\}\right\} = \ sum_ \left\{I\right\} m_ \left\{I\right\} \ vec \left\{v_ \left\{I\right\} ^ \left\{*\right\}\right\} = \ vec \left\{0\right\}$, from where with final the theorem of König relating to the kinetic energy : --> This theorem shows while utilizing the barycentric reference frame (R*) related to the Center of inertia G of the system, and in translatory movement compared to the reference frame of study (R) . He is written:

$E_ \left\{K\right\} = \ frac \left\{1\right\} \left\{2\right\} Mv_ \left\{G\right\} ^ \left\{2\right\} +E_ \left\{K\right\} ^ \left\{*\right\}$.

The kinetic energy of a system is then the sum of two terms: kinetic energy of the Center of mass of (S) affected of its total mass M , $\ frac \left\{1\right\} \left\{2\right\} Mv_ \left\{G\right\} ^ \left\{2\right\}$, and clean kinetic energy of the system in (R*), $E_ \left\{K\right\} ^ \left\{*\right\} \ equiv \ frac \left\{1\right\} \left\{2\right\} \ sum_ \left\{I\right\} m_ \left\{I\right\} v_ \left\{I\right\} ^ \left\{*2\right\}$.

### Application to a solid

A solid is a system of points such as the distances between two unspecified points of (S) are constant. It is about a idealization of a real solid, considered as absolutely rigid.

#### General case: instantaneous axis of rotation

In this case the movement of the solid can be broken up into a movement of sound Center of mass G in (R) and a rotation movement around an instantaneous axis (Δ) in the barycentric reference frame (R*).

More precisely, for a solid one can write the field speeds in the barycentric reference frame (R*) in the form $\ vec \left\{v_ \left\{I\right\} ^ \left\{*\right\}\right\} = \ vec \left\{\ Omega\right\} \ times \ vec \left\{GM_ \left\{I\right\}\right\}$, $\ vec \left\{\ Omega\right\}$ being the instantaneous vector rotation of the solid in (R*) (R), since both [[reference frame (physical)|reference frame] S are in translation]. Its own kinetic energy $E_ \left\{K\right\} ^ \left\{*\right\}$ is expressed then

$E_ \left\{K\right\} ^ \left\{*\right\} = \ frac \left\{1\right\} \left\{2\right\} \ sum_ \left\{I\right\} m_ \left\{I\right\} \ vec \left\{v_ \left\{I\right\} ^ \left\{*\right\}\right\} \ cdot \ left \left(\ vec \left\{\ Omega\right\} \ times \ vec \left\{GM_ \left\{I\right\}\right\} \ right\right) = \ frac \left\{1\right\} \left\{2\right\} \ vec \left\{\ Omega\right\} \ cdot \ left \left(\ sum_ \left\{I\right\} \ vec \left\{GM_ \left\{I\right\}\right\} \ times m_ \left\{I\right\} \ vec \left\{v_ \left\{I\right\} ^ \left\{*\right\}\right\} \ right\right) =\ frac \left\{1\right\} \left\{2\right\} \ vec \left\{L_ \left\{G\right\}\right\} \ cdot \ vec \left\{\ Omega\right\}$,

since $\ vec \left\{L_ \left\{G\right\}\right\} = \ vec \left\{L^ \left\{*\right\}\right\} = \ sum_ \left\{I\right\} \ vec \left\{GM_ \left\{I\right\}\right\} \ times m_ \left\{I\right\} \ vec \left\{v_ \left\{I\right\} ^ \left\{*\right\}\right\}$, Moment kinetic of the solid compared to G, equal to the clean kinetic moment $\ vec \left\{L^ \left\{*\right\}\right\}$ (see theorems of König).

According to the theorem of König, the total kinetic energy of a solid is thus written as follows:

$E_ \left\{K\right\} = \ frac \left\{1\right\} \left\{2\right\} Mv_ \left\{G\right\} ^ \left\{2\right\} + \ frac \left\{1\right\} \left\{2\right\} \ vec \left\{L_ \left\{G\right\}\right\} \ cdot \ vec \left\{\ Omega\right\}$,

that one can regard as the sum of a kinetic energy " of translation" and of a energy kinetic of rotation $E_ \left\{R\right\} \ equiv \ frac \left\{1\right\} \left\{2\right\} \ vec \left\{L_ \left\{G\right\}\right\} \ cdot \ vec \left\{\ Omega\right\}$, so called angular kinetic energy .

#### Case of rotation around a fixed axis

So in addition it rotating around an axis (Δ) fixes there in (R), the kinetic Moment compared to G of the solid is written $\ vec \left\{L_ \left\{G\right\}\right\} =I_ \left\{\ Delta\right\} \ vec \left\{\ Omega\right\}$, where $I_ \left\{\ Delta\right\}$ is the Moment of inertia solid compared to the axis of rotation (Δ). Its kinetic energy of rotation will be put thus in the form:

$E_r = \ begin \left\{matrix\right\} \ frac \left\{1\right\} \left\{2\right\} \ end \left\{matrix\right\} I_ \left\{\ Delta\right\} \ cdot \ omega^2$.

## In relativistic mechanics

In the Theory of relativity of Einstein (used mainly for speeds close to the Speed of light), the kinetic energy is:

$E_c = m c^2 \left(\ gamma - 1\right) = \ gamma m c^2 - m c^2$

$\ gamma = \ frac \left\{1\right\} \left\{\ sqrt \left\{1 - \left(v/c\right)^2\right\}\right\}$

$E_c = \ gamma m c^2 - m c^2$

# \ left (\ frac {1} {\ sqrt {1 - v^2/c^2}} - 1 \ right) m c^2

• Ec the kinetic energy of the body

• v is the speed of the body
• m is its rest mass
• C is the speed of light in the vacuum
• γmc2 is the total energy body
• '' mc2 '' is energy at rest (90 Péta joules per kilogram) expressed in conventional units

The theory of relativity affirms that the kinetic energy of an object tends towards the infinite one when its speed approaches speed of light and that, consequently, it is impossible to accelerate an object until this speed.

One can show that the report/ratio of the relativistic kinetic energy on the Newtonian kinetic energy tends towards 1 when speed v tends towards 0, i.e.,

$\ lim_ \left\{v \ to 0\right\} \left\{\ left \left(\ frac \left\{1\right\} \left\{\ sqrt \left\{1 - v^2/c^2 \\right\}\right\} - 1 \ right\right) m c^2 \ over mv^2/2\right\} =1.$

This result can be obtained by a development limited to the first order of the report/ratio. The term of second order is 0.375 mv4/c ², i.e. for a speed of 10 km/s it is worth 0,04 J/kg, for a speed of 100 km/s it is worth 40 J/kg, etc

When the Gravité is weak and that the object moves at speeds much lower than speed of light (it is the case of the majority of the phenomena observed on Ground), the formula of Newtonian mechanics is an excellent approximation of the relativistic kinetic energy.

## Theorem

This theorem, valid only within the framework of the Newtonian Mechanical , makes it possible to bind the kinetic energy of a system to the work of the force S to which this one is subjected.

### Statement

In a Reference frame galiléen, the variation of the kinetic energy of an object in translation between two points has and B is equal to the algebraic sum of external work of the forces applied to the object between these two points:

$\ Delta E_ \left\{c_ \left\{has \ rightarrow B\right\}\right\} =E_ \left\{c_B\right\} - E_ \left\{c_A\right\} = \ sum \ overline \left\{W_ \left\{F_ \left\{ext_ \left\{has \ rightarrow B\right\}\right\}\right\}\right\}$

--> In a Reference frame galiléen, for a specific body of constant Masse m traversing a way connecting a point has at a point B, the variation of kinetic energy is equal to the sum W work of the external forces which are exerted on the solid in question:

$\ Delta E_ \left\{c_ \left\{AB\right\}\right\} =E_ \left\{c_B\right\} - E_ \left\{c_A\right\} = \ sum W_ \left\{F_ \left\{ext_ \left\{AB\right\}\right\}\right\}$
where EcA and EcB are respectively the kinetic energy of the solid at the points has and B.

### Demonstration

According to the 2 {{E}} law of Newton, the Accélération of the center of gravity is related to the forces which are exerted on the solid by the following relation:
$m \ cdot \ vec \left\{has\right\} = \ vec \left\{F\right\}$
For one length of time dt , the solid moves $\ vec \left\{\right\} = \ vec \left\{v\right\} \ cdot dt$ where $\ vec \left\{v\right\}$ is the speed of the solid. One from of deduced elementary work from the forces:
$\ delta W= \ vec \left\{F\right\} \ cdot \ vec \left\{of the\right\} =m \ cdot \ vec \left\{has\right\} \ cdot \ vec \left\{\right\} =m \ cdot \ frac \left\{\ vec \left\{FD\right\}\right\} \left\{dt\right\} \ cdot \ vec \left\{v\right\} \ cdot dt=m \ cdot \ vec \left\{v\right\} \ cdot \ vec \left\{FD\right\}$
If the solid traverses a way of a point has at a point B, then total work is obtained by making an integral along the way:
$W= \ int_A^ \left\{B\right\} \ vec \left\{F\right\} \ cdot \ vec \left\{of the\right\} = \ int_ \left\{v_A\right\} ^ \left\{v_B\right\} m \ cdot \ vec \left\{v\right\} \ cdot \ vec \left\{FD\right\}$
$\ vec \left\{v\right\} \ cdot \ vec \left\{FD\right\}$ being a total differential , the integral does not depend on the way followed between has and B and can thus be obtained explicitly:
$W=m \ cdot \ int_ \left\{v_A\right\} ^ \left\{v_B\right\} \ vec \left\{v\right\} \ cdot \ vec \left\{FD\right\} = \ frac \left\{1\right\} \left\{2\right\} m \ cdot \ left \left(v_B^2-v_A^2 \ right\right) =E_ \left\{c_B\right\} - E_ \left\{c_A\right\}$

CQFD

## Theorem of the kinetic power

In a reference frame galiléen, the power of the forces applying to the point M is equal to derived compared to time from the kinetic energy.
$P = \ frac \left\{dE_c\right\} \left\{dt\right\} \,$

## Thermal energy as a kinetic energy

The thermal energy is a form of energy due to the total kinetic energy of the Molécule S and Atome S which form the matter. The relation between heat, the Temperature and the kinetic energy of the atoms and the molecules is the object of the Mécanique statistics and the Thermodynamique.

Of quantum nature , the thermal energy transforms into electromagnetic energy by the phenomenon of radiation of the black Corps.

The Chaleur, which represents a thermal energy exchange, is also similar to a work in the direction where it represents a variation of the internal energy of the system. The energy represented by heat directly refers to energy associated with molecular agitation. The conservation of heat and the mechanical energy is the object of the first principle of the Thermodynamique.