Interesting. The video states that there is a point when the transition from acceleration to increased mass occurs, but that is wrong?
So what you're saying is that as you approach c, the mass of the protons increases gradually? Is it exponentially? If so I think by point of transition they mean the part of the exponential curve that transitions from horizontal to vertical. Where the slope of the line at that point would be 45 degrees. Not really a point, but a curve where it transitions fast from one to the other. In terms of time it may occur suddenly to an observer? Or is the video just plain wrong in making that statement?
In the video they state the transition point is at 99.9c where no further energy will be translated into acceleration, but they don't state what happens before 99.9c. I assume it gradually adds to mass more and more and less and less to acceleration. Is there a graph or formula that shows how that change occurs?
There is no transition point - and, yes, the video is not exactly correct when it suggests that there is.
In daily life we are used to the Newtonian principles where acceleration in response to a constant force is constant, regardless of velocity - and the quantitative expression of kinetic energy that results; KE = 0.5 x mv^2.
In relativistic mechanics, kinetic energy increases exponentially as v approaches c. KE = mc² (ˠ-1) where ˠ = 1/sqrt (1-v²/c²
. In other words, as a particle approaches the speed of light, it gets very rapidly more difficult to accelerate - becoming impossible to accelerate to the speed of light; at near light speed, adding/removing energy changes the velocity only extremely slightly.
So what about the mass? Well, relativity makes things a bit awkward, because there are 2 definitions of mass.
One definition is the 'rest mass'. This is the mass of a particle, at rest. This value never changes, as it's definition implies that it was measured at rest (or measured by an observer who is traveling together with the particle). It's the value of 'm' I've used in the above equations.
But what happens if you are not traveling with the particle (you are standing still and a particle whizzes past you)? Or if you have a bunch of particles, which as a whole aren't moving (e.g. a bunch of particles inside a container, but the container isn't moving)? In this definition
mass is nothing more than a reflection of the total energy. In other words, E = mc² - where E is the total energy in the system (the amount of energy you would get from destroying the 'rest mass' + any other other energy that you have added). So if I have a bunch of particles in a box, and I boost their kinetic energy (e.g. by heating). So, if I take a box full of gas, and heat it up, boosting the kinetic energy of the gas molecules, I actually increase the mass of the contents of the box - by virtue of the fact that I have added energy to the contents of the box.
So, if I take a proton which has a rest mass of 1.672 x 10^-27 kg. I could express the mass by its energy equivalence: 1.50 x 10^-10 J (or in more commonly used units 940 MeV). If I now accelerate the proton and add 1.5 x 10^-7 J (940 GeV) of kinetic energy, by the mass-energy equivalence definition above, I can equally well say that the mass of the proton has been increased by 1.67 x 10^-24 kg.
I've drawn a graph of how velocity and kinetic energy ("added mass") go together.
You can see that below about 0.5 c, the KE is almost exactly equal to 0.5 mv². But the curve rapidly steepens as c is approached, where changing velocity even marginally requires vast amounts of energy. However, you'll notice that by approximately 0.9 c, the mass of the particle is approximately doubled (by mass-energy equivalence).
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