# Kinetic Energy

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Kinetic energy represents the energy associated with the motion of an object.[1] It is defined as the work done by a force to accelerate that object from rest to some speed , in the absence of any other forces acting upon the object. Kinetic energy is a scalar and has the same units as work (i.e. Joule).

## Classical mechanics

### Translational kinetic energy

In classical mechanics, the translational kinetic energy of a ridid object, , can be found as:



Where

 is the mass of the object
 is the velocity of the object

### Rotational kinetic energy

The rotational kinetic energy of a rigid object is:



Where

 is the moment of inertia of the object
 is the angular velocity of the object

### Work-Energy theorem

The change of kinetic energy is equal to the total work done on it by the resultant of all forces acting on it. For a point mass this can be expressed as:



Where

 is the initial speed
 is the final speed

Note that if the mass of an object is increased, the increase in kinetic energy increases linearly; if the velocity of an object is increased, the increase in kinetic energy increases quadratically. For example, doubling the mass of an object doubles its kinetic energy; doubling its velocity quadruples its kinetic energy.

### Derivation of translational kinetic energy

The work done by a force accelerating an object from rest, which is the kinetic energy is:



From Newton's second law, the force, , is . Hence we can make the substitution and use the chain rule



This is the same as



In classical mechanics, momentum is given by . Differentiating and substituting into the above equation results in



We want to integrate between 0 and the speed of the object,  as this defines kinetic energy. Performing the integration reveals that the kinetic energy is, as expected, the following:



A similar method may be used to derive the formula for rotational kinetic energy.

## Kinetic Energy in Relativity

In relativity, kinetic energy can be expressed as:



where

 is the Lorentz factor
 is the rest mass
 is the speed of light

This can be shown to be equivalent to the classical equation for kinetic energy, , by performing a binomial expansion on it. Using the result:



Expanding the Lorentz factor in this way, we see:



This simplifies to



For speeds encountered everyday, which are a lot less than that of [[speed of light|light] (such that ), all terms apart from the first are very small and can be neglected. Hence, the formula reduces to:



which is the classical formula.

### Derivation

The kinetic energy is the work done accelerating a particle from rest to some speed . Suppose the particle is at rest at  and speed  at position . Hence:



Since , and a perpendicular force does no work, we can ignore the perpendicular component and write:



Since , we find the integral can be rewritten as:



where  is the initial speed and hence 0 by definition and  is the final speed.

Performing the integration reveals the kinetic energy as:



## References

1. Serway and Beichner, Physics for Scientists and Engineers, Fifth Edition