The kinetic energy of an object is the extra 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 current velocity . Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes. Negative work of the same magnitude would be required to return the body to a state of rest from that velocity.
The kinetic energy of a single object is completely frame-dependent (relative). For example, a bullet racing by a non-moving observer has kinetic energy in the reference frame of this observer, but the same bullet has zero kinetic energy in the reference frame which moves with the bullet. The kinetic energy of systems of objects, however, may sometimes not be completely removable by simple choice of reference frame. When this is the case, a residual minimum kinetic energy remains in the system as seen by all observers, and this kinetic energy (if present) contributes to the system's invariant mass, which is seen as the same value in all reference frames, and by all observers.
History and Etymology
The adjective "kinetic" has its roots in the Greek word κίνηση (kinesis) meaning "motion" – the same root as in the word cinema (referring to motion pictures).
The principle in classical mechanics that E ∝ mv² was first theorized by Gottfried Leibniz and Johann Bernoulli, who described kinetic energy as the "living force", vis viva . Willem 's Gravesande of the Netherlands provided experimental evidence of this relationship. By dropping weights from different heights into a block of clay, 's Gravesande determined that their penetration depth was proportional to the square of their impact speed. Émilie du Châtelet recognized the implications of the experiment and published an explanation.
The terms "kinetic energy" and "work" with their present scientific meanings date back to the mid 19th century. Early understandings of these ideas can be attributed to Gaspard-Gustave Coriolis who in 1829 published the paper titled Du Calcul de l'Effet des Machines outlining the mathematics of kinetic energy. William Thomson, later Lord Kelvin, is given the credit for coining the term "kinetic energy" c. 1849 - 1851.
Introduction
Main article: EnergyThere are various forms of energy: chemical energy, heat, electromagnetic radiation, potential energy (gravitational, electric, elastic, etc.), nuclear energy, rest energy. These can be categorized in two main classes: potential energy and kinetic energy.
Kinetic energy can be best understood by examples that demonstrate how it is transformed to and from other forms of energy. For example, a cyclist will use chemical energy that was provided by food to accelerate a bicycle to a chosen speed. This speed can be maintained without further work, except to overcome air-resistance and friction. The energy has been converted into kinetic energy – the energy of motion – but the process is not completely efficient and heat is also produced within the cyclist.
The kinetic energy in the moving cyclist and the bicycle can be converted to other forms. For example, the cyclist could encounter a hill just high enough to coast up, so that the bicycle comes to a complete halt at the top. The kinetic energy has now largely been converted to gravitational potential energy that can be released by freewheeling down the other side of the hill. (Since the bicycle lost some of its energy to friction, it will never regain all of its speed without further pedaling. Note that the energy is not destroyed; it has only been converted to another form by friction.) Alternatively the cyclist could connect a dynamo to one of the wheels and also generate some electrical energy on the descent. The bicycle would be traveling more slowly at the bottom of the hill because some of the energy has been diverted into making electrical power. Another possibility would be for the cyclist to apply the brakes, in which case the kinetic energy would be dissipated through friction as thermal energy.
Like any physical quantity which is a function of velocity, the kinetic energy of an object depends on the relationship between the object and the observer's frame of reference. Thus the kinetic energy of an object is not invariant.
Examples
Spacecraft use chemical energy to take off and gain considerable kinetic energy to reach orbital velocity. This kinetic energy gained during launch will remain constant while in orbit because there is almost no friction. However it becomes apparent at re-entry when the kinetic energy is converted to heat.
Kinetic energy can be passed from one object to another. In the game of billiards, the player gives kinetic energy to the cue ball by striking it with the cue stick. If the cue ball collides with another ball, it will slow down dramatically and the ball it collided with will accelerate to a speed as the kinetic energy is passed on to it. Collisions in billiards are effectively elastic collisions, where (by definition) kinetic energy is preserved. In inelastic collisions, kinetic energy is dissipated as: heat, sound, binding energy (breaking bound structures), or other kinds of energy.
Flywheels are being developed as a method of energy storage (see Flywheel energy storage). This illustrates that kinetic energy can also be rotational.
Calculations
There are several different equations that may be used to calculate the kinetic energy of an object. In many cases they give almost the same answer to well within measurable accuracy. Where they differ, the choice of which to use is determined by the velocity of the body or its size. Thus, if the object is moving at a velocity much smaller than the speed of light, the Newtonian (classical) mechanics will be sufficiently accurate; but if the speed is comparable to the speed of light, relativity starts to make significant differences to the result and should be used. If the size of the object is sub-atomic, the quantum mechanical equation is most appropriate.
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
where
is the mass and
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 traveling at 18 meters per second (40 mph) as
Note that the kinetic energy increases with the square of the speed. This means, for example, that an object traveling twice as fast will have four times as much kinetic energy. As a result of this, a car traveling twice as fast requires four times as much distance to stop (assuming a constant braking force. See mechanical work).
The kinetic energy of an object is related to its momentum by the equation:
where:
For the translational kinetic energy, that is the kinetic energy associated with rectilinear motion, of a body with constant mass
where:
The kinetic energy of any entity is unique to the reference frame in which it is measured. An isolated system is one for which energy can neither enter nor leave, and has a total energy which is unchanging over time as measured in any reference frame. Thus, the chemical energy converted to kinetic energy by a rocket engine will be 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 energy, etc) will be conserved over time, regardless of the choice of reference frame. However, different observers moving with different reference frames will disagree on the value of this conserved energy.
In addition, although the energy of such systems is dependent on the choice of reference frame, the minimal total energy which is seen in any frame will be the total energy seen by observers in the center of momentum frame; this minimal energy corresponds to the invariant mass of the aggregate. The calculated value of this invariant mass compensates for changing energy in different frames, and is thus the same for all frames and observers.
Derivation
The work done accelerating a particle during the infinitesimal time interval dt is given by the dot product of force and displacement :
Applying the product rule we see that:
Therefore (assuming constant mass), the following can be seen:
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:
This equation states t
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