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All Basic Mathematics Math Formulas:. Algebra Formulas. Circumcenter Formula. Mean Median Mode Formula. Square Root Formula. Pythagorean Theorem Formula. 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. Different observers moving with different reference frames would however disagree on the value of this conserved energy.

The kinetic energy of such systems Math Ke Formula Uk depends on the choice of reference frame: the reference frame that gives the minimum value of Uk Ke Math Formula that energy is the center of momentum frame, i.

This minimum kinetic energy contributes to the invariant Ke Uk Math Formula Uk Formula Math Ke mass of the system as a whole. The work done in accelerating a particle with mass m during the infinitesimal time interval dt is given by the dot product of force F and the infinitesimal displacement dx.

However, also see the special relativistic derivation below. Since this is a Math Ke Formula Uk total differential that is, it only depends on the final state, not how the particle got Ke Formula Uk Math there , we can integrate it and call the result kinetic energy.

Assuming the object was at rest at time 0, we integrate from time 0 to time t because the work done by Math Ke Formula Uk the force to bring the object from rest to velocity v is equal to the work necessary to do the reverse:.

This equation states that the kinetic energy E k is equal to Math Ke Formula Uk the integral of the dot product of the velocity v of a body and the infinitesimal change of the body's momentum p.

It is assumed that the body starts with no kinetic Math Ke Formula Uk Math Ke Formula Uk energy when it is at rest motionless. A system of bodies may have internal kinetic energy due Math Ke Formula Uk Math Ke Formula Uk to the relative motion of the bodies in the system. For example, in the Solar System the planets and planetoids are orbiting the Sun.

In a tank of gas, the molecules are moving Math Ke Formula Uk in all directions. The kinetic energy of the system is the sum of the kinetic energies of the bodies it contains. A macroscopic body that is stationary i. These all contribute to the body's mass, as provided by the special theory of relativity.

When discussing movements of a macroscopic body, the kinetic energy referred to is usually that of the macroscopic movement only. However Math Ke Formula Uk all internal energies of all types contribute to body's mass, inertia, and total energy. In fluid Math Ke Formula Uk dynamics , the kinetic energy per unit volume at each point in an incompressible fluid flow field is called the dynamic pressure at that point. The speed, and thus the kinetic energy of a Ke Formula Math Uk single object is frame-dependent relative : it can take any non-negative value, by choosing a suitable inertial Math Ke Formula Uk Math Ke Formula Uk frame of reference.

For example, a bullet passing an observer has kinetic energy in the reference frame Math Uk Ke Formula of this observer. The same bullet is stationary to an observer moving with the same velocity as the bullet, and so has zero kinetic energy.

In any other case, the total 8th Standard Maths Algebra Formulas 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 Ke Uk Formula Math independent of the reference frame. The total kinetic energy of a system depends on the inertial frame of reference : it is the sum of the total kinetic energy in a center of momentum frame and the kinetic energy the total mass would have if it were concentrated in the center of mass.

Substituting, we get: [9]. Thus the kinetic energy of a system is lowest to center of momentum reference frames, i. In any different frame of reference, there is additional kinetic energy corresponding to the total mass moving at the speed of the center of mass. The kinetic Math Ke Formula Uk energy of the system in the center of momentum frame is a quantity that is invariant all observers see it to be the same.

It sometimes is convenient to split the total kinetic Math Ke Formula Uk energy of a body into the sum of the body's center-of-mass translational kinetic energy and Math Ke Formula Uk Math Ke Formula Uk Math Ke Formula Uk the energy of rotation around the center of mass rotational energy :.

Thus the kinetic energy of a tennis ball in flight is the kinetic energy due to its rotation, plus the kinetic energy Math Ke Formula Uk due to its translation. If a body's speed is a significant fraction of the speed of light , it is necessary to use relativistic mechanics to calculate its kinetic energy. In special relativity theory, the expression for linear momentum is modified. Integrating by parts yields.

This formula shows that the work expended accelerating an object from rest approaches infinity as the velocity approaches the speed of Math Ke Formula Uk light. Thus it is impossible to accelerate an object across this boundary. The mathematical by-product of this calculation is the mass-energy equivalence formula�the body at rest must have energy content. This is done Math Ke Formula Uk by binomial approximation or by taking the first two terms of the Taylor expansion for the Math Ke Formula Uk Math Ke Formula Uk Math Ke Formula Uk reciprocal square root:.

When objects move at a speed much slower than light e. The next term in the Taylor series approximation. This can also be expanded as a Taylor series , the first Math Ke Formula Uk Math Ke Formula Uk term of which is the simple expression from Newtonian mechanics: [10]. This suggests that the formulae for Math Ke Formula Uk energy and momentum are not special and axiomatic, but concepts emerging from the equivalence of mass and energy and the principles of relativity.

Consider the case of a metric that is diagonal and Ke Math Formula Uk Math Ke Formula Uk spatially isotropic g tt , g ss , g ss , g ss. This means clocks run slower and measuring rods are shorter near massive bodies. In quantum mechanics , observables like kinetic energy are represented as Uk Ke Math Formula operators. The kinetic energy operator in the non-relativistic case can be written as.

The density functional formalism of quantum mechanics requires knowledge of the electron density only , i. From Wikipedia, the free encyclopedia.