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Tony Batchelo
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we all accept the E=mc2 formula but what is the mathmatical proof for the c2, why for example is it not c3 or c4?
Many of us do not just accept it but know its derivation- it can be found in any introductory text on relativity- and so understand why it is [itex]c^2[/itex]. I see Keven Axion has given a link to the derivation.Tony Batchelo said:we all accept the E=mc2 formula but what is the mathmatical proof for the c2, why for example is it not c3 or c4?
DaleSpam said:Let's say that we begin with the supposition that mass has some intrinsic amount of energy, even when it is at rest, and we want to find the formula that determines how much energy is in a given amount of mass. First, we know that the units of mass and energy differ by a speed² so we can immediately write E=kmv².
So, what v can we use? The mass is at rest (v=0). So we cannot use its speed; it must be some sort of universal constant with units of speed. The only such constant is the speed of light, so we know E=kmc² just from consideration of units.
Determining that k=1 requires some actual derivation, so if anything you should ask why k=1 instead of k=1/2. But c² should be obvious: what else could it possibly be?
But your answer doesn't really make sense. Du Chatelet showed that the object's kinetic energy is proportional to its speed squared. But mc^2 is not a kinetic energy; c is not the speed of the object.jonlg_uk said:I ANSWERED THIS QUESTION GIVING A FULL HISTORY OF WHERE c^2 arrived from...
jonlg_uk said:The answer lies with the discovery made by a brilliant French aristocrat, Emilie du Chatelet.
At the age of 23, du Chatelet discovered a talent for advanced mathematics which she relished. So much so that she began to formulate ideas of her own; ideas that challenged the great physicists, including Sir Isaac Newton.
Newton stated that the energy (or force) of a moving object could simply be expressed as its mass multiplied by its velocity. But while corresponding with a German scientist called Gottfried Leibniz, du Chatelet learned that Leibniz considered the energy of a moving object is better described if its velocity is squared. But how to test this? Du Chatelet tried an experiment that would prove her point – dropping lead balls into clay.
Newton's formula says that doubling the velocity of a ball would double its energy and so one would expect it to travel twice as far into the clay. But if the velocity is squared, as Leibniz and du Chatelet believed, the force should be four times greater, and the ball should travel four times the distance into the clay.
Du Chatelet conducted her lead ball experiment and sure enough, doubling the velocity of the ball (by dropping it from twice the height) resulted in the ball traveling four times further into the clay. This simple but brilliant experiment proved that when calculating the energy of moving objects, the velocity at which they travel must be squared. The energy of an object is a function of its velocity squared – it is for this reason that the speed of light in Einstein's equation must be squared.
This was a factor that profoundly changed the meaning of Einstein's equation – since c is already a large number, once squared it is vast. Thus, a vast amount of energy (E) can be associated with a very small amount of mass (m) because mass is always multiplied by the speed of light (c) squared – a vast number. Under these laws, even a tiny amount of mass will equate to a huge amount of energy.
jonlg_uk said:The answer lies with the discovery made by a brilliant French aristocrat, Emilie du Chatelet.
At the age of 23, du Chatelet discovered a talent for advanced mathematics which she relished. So much so that she began to formulate ideas of her own; ideas that challenged the great physicists, including Sir Isaac Newton.
Newton stated that the energy (or force) of a moving object could simply be expressed as its mass multiplied by its velocity. But while corresponding with a German scientist called Gottfried Leibniz, du Chatelet learned that Leibniz considered the energy of a moving object is better described if its velocity is squared. But how to test this? Du Chatelet tried an experiment that would prove her point – dropping lead balls into clay.
Newton's formula says that doubling the velocity of a ball would double its energy and so one would expect it to travel twice as far into the clay. But if the velocity is squared, as Leibniz and du Chatelet believed, the force should be four times greater, and the ball should travel four times the distance into the clay.
Du Chatelet conducted her lead ball experiment and sure enough, doubling the velocity of the ball (by dropping it from twice the height) resulted in the ball traveling four times further into the clay. This simple but brilliant experiment proved that when calculating the energy of moving objects, the velocity at which they travel must be squared. The energy of an object is a function of its velocity squared – it is for this reason that the speed of light in Einstein's equation must be squared.
This was a factor that profoundly changed the meaning of Einstein's equation – since c is already a large number, once squared it is vast. Thus, a vast amount of energy (E) can be associated with a very small amount of mass (m) because mass is always multiplied by the speed of light (c) squared – a vast number. Under these laws, even a tiny amount of mass will equate to a huge amount of energy.
At the age of 23, du Chatelet discovered a talent for advanced mathematics which she relished. So much so that she began to formulate ideas of her own; ideas that challenged the great physicists, including Sir Isaac Newton. Newton stated that the energy (or force) of a moving object could simply be expressed as its mass multiplied by its velocity. But while corresponding with a German scientist called Gottfried Leibniz, du Chatelet learned that Leibniz considered the energy of a moving object is better described if its velocity is squared. But how to test this? Du Chatelet tried an experiment that would prove her point ? dropping lead balls into clay.
Du Chatelet conducted her lead ball experiment and sure enough, doubling the velocity of the ball (by dropping it from twice the height) resulted in the ball traveling four times further into the clay. This simple but brilliant experiment proved that when calculating the energy of moving objects, the velocity at which they travel must be squared. The energy of an object is a function of its velocity squared ? it is for this reason that the speed of light in Einstein's equation must be squared.
The problem is that the above bolded is a bunch of crap. The main offenders being the equivalence between force and energy and the supposition that 0.5mv^2 somehow means that people were on the right track to get mc^2. It ignores the fact that the relationship between kinetic energy and the square of velocity has been standard mechanics for centuries and that the relationship of mc^2 is completely different. E=mc^2 is the rest energy and is not part of the kinematics, as a constant offset it does not affect any classical mechanics. If we take the energy relation for a moving object, we can get the Newtonian physics energy relationship back out by a simple Taylor's expansion. The first term is the rest energy and the second is the classical kinetic energy.
So for classical kinematics, relativity does not change anything, in fact, it just reaffirms the original equations.
Looking at what they said about Newton's statement, I just got my requested copy of the Principia and while I am still going through the notes I believe that the statement is more of a misinterpretation. The Principia suffers from two problems. The first is that it is in Latin, the second is that it is largely devoid of equations, Newton talked out the relationships and both of these aspects can introduce gross misinterpretations. For example, the Cohen and Whitman translations specifies Newton's Second Law (Book 1 Law 2) as
A change in motion is proportional to the motive force impressed and takes place along the straight line in which that force is impressed.
If some force generates any motion, twice the force will generate twice the motion, and three times the force will generate three times the motion, whether the force is impressed all at once or successively by degrees.
This is open to a lot of interpretation even after its translation into English. What is meant by motion? An incorrect interpretation would think that it is velocity, thus Newton is saying that a force generates a velocity (ignoring his stipulation of change in motion in the heading for the law). Twice the force will generate twice the velocity and so on. Add in the incorrect assumption that force and energy are the same (which I do not think Newton supposed) and you get that Newton stated Force is proportional to Velocity. However, Newton is talking about a change in motion being proportional to the force. Newton's "motion" here is momentum. He is saying that the change in momentum (because he is talking about an impulse force here, not a continuous foce) is proportional to the impulse force that is applied. In addition, if we apply twice the impulse force, we get twice the change in momentum. Or, if we successively apply an impulse force we will successively add up the change in momentum. This is true and agrees with modern classical mechanics. Newton is saying that Force is proportional to the change in Mass*Velocity.
Another point is that the Principia was originally written around the idea of explaining the motion of the heavenly bodies, to put it poetically. The concept of kinetic energy was not developed here. So it is certainly correct to say that Leibnitz and others developed the idea of energy. However, the translators of my copy note that Newton did derive a value that would be equivalent to energy and his relationship is that it is equal to 0.5mv^2.
E= Mx(C Squared) by why squared is a famous equation in physics, also known as the mass-energy equivalence equation. It states that the energy of a body (E) is equal to its mass (M) multiplied by the speed of light (C) squared.
The equation was first proposed by Albert Einstein in 1905 as part of his theory of special relativity. However, it was not until 1915, with the development of his theory of general relativity, that the equation gained widespread recognition.
E= Mx(C Squared) by why squared is the foundation of the theory behind nuclear energy. It explains how a small amount of mass can be converted into a large amount of energy, as demonstrated by nuclear reactions such as fission and fusion.
Yes, E= Mx(C Squared) by why squared is a universally applicable equation. It applies to all forms of energy and matter, regardless of their size or speed. However, it is most commonly used in the study of subatomic particles and the behavior of objects at high speeds.
Yes, E= Mx(C Squared) by why squared can be derived from other equations in physics, such as the Lorentz transformation and the conservation of energy. However, it is also a fundamental equation in its own right and is often used to derive other equations in the field of physics.