# Kinetic energy

The kinetic energy of an object is the energy which it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes. The same amount of work is done by the body in decelerating from its current speed to a state of rest.

The speed, and thus the kinetic energy of a single object is frame-dependent (relative): it can take any non-negative value, by choosing a suitable inertial frame of reference. For example, a bullet passing an observer has kinetic energy in the reference frame of this observer. The same bullet is stationary from the point of view of an observer moving with the same velocity as the bullet, and so has zero kinetic energy. By contrast, the total kinetic energy of a system of objects cannot be reduced to zero by a suitable choice of the inertial reference frame, unless all the objects have the same velocity. In any other case the total kinetic energy has a non-zero minimum, as no inertial reference frame can be chosen in which all the objects are stationary. This minimum kinetic energy contributes to the system's invariant mass, which is independent of the reference frame.

In classical mechanics, the kinetic energy of a non-rotating object of mass m traveling at a speed v is ½ mv². Inrelativistic mechanics, this is only a good approximation when v is much less than the speed of light

## Newtonian kinetic energy

### Kinetic energy of rigid bodies

In classical mechanics, the kinetic energy of a point object (an object so small that its mass can be assumed to exist at one point), or a non-rotating rigid body, is given by the equation

$E_k = frac{1}{2} mv^2$

where $m$ is the mass and $v$ is the speed (or the velocity) of the body. In SI units (used for most modern scientific work), mass is measured in kilograms, speed in metres per second, and the resulting kinetic energy is in joules.

For example, one would calculate the kinetic energy of an 80 kg mass (about 180 lbs) traveling at 18 metres per second (about 40 mph, or 65 km/h) as

Ek = (1/2) · 80 · 182 J = 12.96 kJ

Since the kinetic energy increases with the square of the speed, an object doubling its speed has four times as much kinetic energy. For example, a car traveling twice as fast as another requires four times as much distance to stop, assuming a constant braking force.

The kinetic energy of an object is related to its momentum by the equation:

$E_k = frac{p^2}{2m}$

where:

$p;$ is momentum
$m;$ is mass of the body

For the translational kinetic energy, that is the kinetic energy associated with rectilinear motion, of a rigid body with constant mass $m;$, whose center of mass is moving in a straight line with speed $v;$, as seen above is equal to

$E_t = frac{1}{2} mv^2$

where:

$m;$ is the mass of the body
$v;$ is the speed of the center of mass of the body.

The kinetic energy of any entity depends on the reference frame in which it is measured. However the total energy of an isolated system, i.e. one which energy can neither enter nor leave, does not change in whatever reference frame it is measured. Thus, the chemical energy converted to kinetic energy by a rocket engine is divided differently between the rocket ship and its exhaust stream depending upon the chosen reference frame. This is called the Oberth effect. But the total energy of the system, including kinetic energy, fuel chemical energy, heat, etc., is conserved over time, regardless of the choice of reference frame. Different observers moving with different reference frames disagree on the value of this conserved energy.

The kinetic energy of such systems depends on the choice of reference frame: the reference frame that gives the minimum value of that energy is the center of momentum frame, i.e. the reference frame in which the total momentum of the system is zero. This minimum kinetic energy contributes to the invariant mass of the system as a whole.

#### Derivation

The work done accelerating a particle during the infinitesimal time interval dt is given by the dot product of force and displacement:

$mathbf{F} cdot d mathbf{x} = mathbf{F} cdot mathbf{v} d t = frac{d mathbf{p}}{d t} cdot mathbf{v} d t = mathbf{v} cdot d mathbf{p} = mathbf{v} cdot d (m mathbf{v}),,$

where we have assumed the relationship p = m v. (However, also see the special relativistic derivation below.)

Applying the product rule we see that:

$d(mathbf{v} cdot mathbf{v}) = (d mathbf{v}) cdot mathbf{v} + mathbf{v} cdot (d mathbf{v}) = 2(mathbf{v} cdot dmathbf{v}).$

Therefore (assuming constant mass), the following can be seen:

$mathbf{v} cdot d (m mathbf{v}) = frac{m}{2} d (mathbf{v} cdot mathbf{v}) = frac{m}{2} d v^2 = d left(frac{m v^2}{2} ight).$

Since this is a total differential (that is, it only depends on the final state, not how the particle got there), we can integrate it and call the result kinetic energy:

$E_k = int mathbf{F} cdot d mathbf{x} = int mathbf{v} cdot d (m mathbf{v}) = int d left(frac{m v^2}{2} ight) = frac{m v^2}{2}.$

This equation states that the kinetic energy (Ek) is equal to the integral of the dot product of the velocity (v) of a body and the infinitesimal change of the body'smomentum (p). It is assumed that the body starts with no kinetic energy when it is at rest (motionless).

### Rotating bodies

If a rigid body is rotating about any line through the center of mass then it has rotational kinetic energy ($E_r,$) which is simply the sum of the kinetic energies of its moving parts, and is thus given by:

$E_r = int frac{v^2 dm}{2} = int frac{(r omega)^2 dm}{2} = frac{omega^2}{2} int{r^2}dm = frac{omega^2}{2} I = egin{matrix} frac{1}{2} end{matrix} I omega^2$

where:

• ω is the body's angular velocity
• r is the distance of any mass dm from that line
• $I,$ is the body's moment of inertia, equal to $int{r^2}dm$.

(In this equation the moment of inertia must be taken about an axis through the center of mass and the rotation measured by ω must be around that axis; more general equations exist for systems where the object is subject to wobble due to its eccentric shape).

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