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).
Translational kinetic energy
In classical mechanics, the translational kinetic energy of a ridid object, , can be found as:
Rotational kinetic energy
The rotational kinetic energy of a rigid object is:
- is the moment of inertia of the object
- is the angular velocity of the object
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
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:
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.
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:
- Serway and Beichner, Physics for Scientists and Engineers, Fifth Edition