# Kinetic Energy

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Kinetic energy represents the energy associated with the motion of an object. 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 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: 