The idea of kinetic energy originates from the era of Newton. It took several decades before physicists could even agree on what kinetic energy was. But that's all history and there is no longer any debate on the topic.
Kinetic energy is the energy of motion. That motion could be moving like an arrow in flight,or it could be rotation, or vibration. However, I want to talk about just simple "translational motion".The equation for kinetic energy is mathematically straightforward. If you have a mass m, moving at velocity v, kinetic energy is simply one half times the mass times the velocity squared - KE And you can use that equation to solve all sorts of introductory physics problems.
KE = 1/2 mv²
It's a great tool and it served scientists well from the early 1700s until about 1905, until came in the picture.
Einstein was interested in the laws of physics when things traveled at near the speed of light. He came up with an equation not for kinetic energy, but for the entire energy of an object.
I specially sense you are thinking about his famous equation E = MC².
But that's not what I am talking about, while this equation (E = MC²) on an special case, only if an object is at rest. It's the energy associated with the stationary mass of the object.
Einstein's physics is to talk about a moving object, you need to use a slightly different equation. For a moving object, Einstein found that energy is equal to a thing called gamma times mc².
E = ymc² ----- (1)
Where;
E=Energy
Y=Gamma
C= Speed of light squared
Also,
Gamma is equal to one over the square root of one minus the velocity of the object squared divided by the speed of light squared.
The energy E of a moving object is simply gamma m c squared. The total energy of a body that isn't acting under a force is equal to the kinetic energy plus the mass energy. So, we can do some simple math and we find that, at least according to Einstein, the kinetic energy of an object would be the quantity gamma minus one times m c squared. Represented in the picture below
I guess the first thing to do is to see if Einstein's formula for kinetic energy and the classical formula look the same. I have plotted both of them here.
We see that they disagree for velocities that are a large fraction of the speed of light. However, as we look at lower speeds - say under ten percent the speed of light, the two graphs And it's important to remember that 10% the speed of light is just slightly slower than going once around the Earth in a single second.
So, this means that the classical formula for kinetic energy is pretty good for any reasonable speed you might encounter. Of course, what I've shown you here is simply a graph of both ways to calculate kinetic energy.
How about showing how one can get from Einstein's equation to the classical equation?How do you do that? Well, for this, we need to use some mathematical tricks, specifically one called the binomial expansion. This mathematical equation was discovered in about 1655 by Isaac Newton,cheap although it wasn't formally proved until 1736 by mathematician John Colson who held the same Lucasian Chair of Mathematics at the university of Cambridge that Isaac Newton held before him.
The binomial expansion says that the expression one plus x raised to the n power can be written as an infinite sum of the form 1 plus n times x plus n times (n minus 1) times x squared, divided by 2 factorial plus n times (n minus one) times (n minus two) times x cubed, divided by 3 factorial and so on...
This equation works for any n and for all x's that are between minus 1 and plus 1. If you want to work out the derivation, well that's up to you. But that's how it goes. So we can look at that gamma factor of Einstein and see how the binomial expansion can help us.
One over the square root of one minus v squared over c squared can be written as one minus v squared divided by c squared to the minus one half power.
So, since v divided by c is less than one, that means that v squared divided by c squared is definitely less than one. And this means we can use the binomial expansion.
We plop in minus one half for n and minus v squared over c squared for x and do some simplification and we end up with the final answer. This is for gamma.
Remember that gamma times mc squared is the total energy. So, we multiply this series of terms with the mc squared and we get this expression for energy,
which is m c squared plus (one half m v squared) plus (three eighths m v to the fourth, divided by c squared) and so on.
Remember that, in relativity, energy is the rest mass energy plus the kinetic energy and that's what we see here. The m c squared is the rest mass energy and the rest is the kinetic energy.
Thus, to find the kinetic energy, we subtract off the m c squared, and we're left with just kinetic energy. And, as we expected, it starts out with the one half m v squared and then has some other terms.
So now you might ask yourself just how big of a deal are these extra terms?
We can kind of get a sense of that by asking at what velocity the second term is big enough to change the kinetic energy by a one percent difference from you get from the classical one half m v squared.
When you do that, you find out that you have to be going at about twelve percent the speed of light, which is about twenty two thousand miles per second, or about thirty five thousand kilometers per second. Ballpark, that's about fast enough to circle the globe once in a single second. Given that the fastest object made by mankind is the Sparker Solar Probe, which travels at a speed of about seven miles per second or just over eleven kilometers per second, we see why physics professors only teach the classical equation for kinetic energy, which I remind you, is one half m v².
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