# Kinetic Energy approximation both Einstein's and Newton's

• TWest
In summary, at low energies we use the Newtonian definition of kinetic energy which is Ek = 1/2MV2 Now at high energies/high volecities it turns into the relativity version of it which is Ek = MC2G-MC2. Now my question is at what point does this change from Newton's to Einstein's? Secondly, can you point me in the direction of a model that is accurate at low velocity and high?The Einstein equation is accurate. The Newtonian one is inaccurate. It's just that at higher and higher energies, the Newtonian expression gets more and more inaccurate.
TWest
Something has been bothering me for quite some time and when you cannot figure it out yourself ask a higher power, which happens to be this forum.

Basically, at low energies we use the Newtonian definition of kinetic energy which is Ek = 1/2MV2 Now at high energies/high volecities it turns into the relativity version of it which is Ek = MC2G-MC2. Now my question is at what point does this change from Newton's to Einstein's? Secondly, can you point me in the direction of a model that is accurate at low velocity and high?

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The Einstein version is good for all speeds. It just depends on how accurate you need to be.

There's no point where one formula turns into another. The Einstein equation is accurate. The Newtonian one is inaccurate. It's just that at higher and higher energies, the Newtonian expression gets more and more inaccurate. For example at a low energy, the Newtonian expression may be off by only .05%, while at a higher energy, it may be off by 50%, or 100% or more. As you get closer and closer to the speed of light, the inaccuracy becomes more and more extreme.

There's no point where one equation turns into the other though.

If you take the formula for the total energy of a moving mass:

$$E_t = \frac{mc^2}{sqrt{1-\frac{v^2}{c^2}}}$$

and expand it into a series, you get :

$$E_t = mc^2+ \frac{mv^2}{2}+ m \frac{3v^4}{8c^2}...$$

The first term is the Energy equivalence of the rest mass.

The second term is the Newtonian definition of KE.

And the additional terms make up the KE difference due to relativity.

As to when to use Newton vs. Einstein:
Einstein's formula alwaysgive the more accurate answer for any value of v.
It really just depends on how accurate you need your answer to be for the particular problem at hand as to whether or not you can get away with using Newton's formula.

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TWest said:
Something has been bothering me for quite some time and when you cannot figure it out yourself ask a higher power, which happens to be this forum.

Basically, at low energies we use the Newtonian definition of kinetic energy which is Ek = 1/2MV2 Now at high energies/high volecities it turns into the relativity version of it which is Ek = MC2G-MC2. Now my question is at what point does this change from Newton's to Einstein's? Secondly, can you point me in the direction of a model that is accurate at low velocity and high?

Weidner And Sells Elementary Modern Physics gives the guide that that calculations using Ek=mv2/2 will be accurate within 1% when v/s < 0.1.

TWest said:
Basically, at low energies we use the Newtonian definition of kinetic energy which is Ek = 1/2MV2 Now at high energies/high volecities it turns into the relativity version of it which is Ek = MC2G-MC2. Now my question is at what point does this change from Newton's to Einstein's? Secondly, can you point me in the direction of a model that is accurate at low velocity and high?

Einstein's equation:

$$E_k = \frac{m c^2}{1-v^2/c^2} - mc^2$$

is accurate at low velocity and high.

Newtons equation is an approximation which is fairly accurate at low velocities. At 0.2c there is error of 2% in Newtons equation and the error gets much larger as velocities get nearer c.

Technically, Newton's equation is inaccurate for any velocity greater than zero.

BarCouSeH
Newton's equation of kinetic energy is as accurate as Einstein's one at any velocity, if you add the increasing mass of the moving objet into Newton's equation.

eljaala said:
Newton's equation of kinetic energy is as accurate as Einstein's one at any velocity, if you add the increasing mass of the moving objet into Newton's equation.

This is simply wrong and is a classic example of the pitfalls of the obsolete notion of relativistic mass. The relativistic mass is: mγ. Putting into Newton's kinetic energy you get:

(1/2) m γ v^2

compare to the correct:

(γ - 1) mc^2

Noting that γ = 1/ sqrt(1-v^2/c^2), algebra will show you these are not the same. You can also plug numbers into see that they are not the same.

The only common formula for which relativistic mass 'works' is momentum. It fails for force, it fails for kinetic energy.

[edit: if you take the series expansion of the wrong and right formulas above, the first order difference 1/8 v^4/c^2. That is, they differ in even the first correction to Newtonian formula.]

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## 1. What is kinetic energy?

Kinetic energy is the energy an object possesses due to its motion. It is a type of energy that is associated with the movement of an object.

## 2. How is kinetic energy calculated using Newton's equation?

According to Newton's second law of motion, the kinetic energy of an object can be calculated by multiplying its mass by the square of its velocity and dividing by two: KE = 1/2 * m * v^2.

## 3. How does Einstein's theory of relativity affect our understanding of kinetic energy?

Einstein's theory of relativity states that energy and mass are interchangeable, and that the speed of light is constant. This means that as an object's velocity approaches the speed of light, its mass and energy increase. Therefore, the traditional equation for kinetic energy is no longer accurate at high speeds and must be replaced with the relativistic equation: KE = (mc^2) / sqrt(1-(v^2/c^2)).

## 4. Is kinetic energy conserved in collisions?

According to the law of conservation of energy, energy cannot be created or destroyed, only transferred or transformed. Therefore, kinetic energy is conserved in collisions, meaning the total kinetic energy before a collision is equal to the total kinetic energy after the collision.

## 5. Can kinetic energy be negative?

Yes, kinetic energy can be negative. This occurs when an object is decelerating or moving in the opposite direction of its initial velocity. In these cases, the kinetic energy is represented as a negative value in the equations.

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