9.2 Energy Conservation

Key Takeaways

welcome back to age 20 special relativity in this section we're going to further investigate the energy momentum 4 vector which we introduced in the previous sections but here we focus on the zeroest component the first component of this vector where we find mass m a for particle a times the proper velocity zeroth component is equal to m a times c times one over one minus u a squared over c square which is the energy of this particle a over c or in other words the energy is equal to the mass times c square over one square root of one minus u a squared c squared so let's discuss or look at let's have a look at this um a little bit more the first question we can ask how does this now look like for particles which travel with a reasonably low velocity so ua much smaller than c so we can tailor expand this um following this equation here which we discussed earlier um and we find that the energy is equal to mac square that's the first term which we call rest mass the energy given adjusts the rest mass by the mass of the particle times c square plus one half m a c squared times u a squared over c squared the c squares cancel and we find what we known as the kinetic energy one half m v square or in this case one half m a u a square that looks very familiar so the energy of a particle is given by its rest mass plus its kinetic energy all right um now investigating this four vector then we can ask you know what the how does the invariant interval look like how does this property which is inverant and the laurel's transformation look like if we multiply the vector with itself here we find minus e square over c square plus the three momentum squared is equal to minus m0 c square or in other words we find this energy momentum energy mass relation energy momentum mass relation where the energy is given by the momentum square times c square plus the rest mass square times c to the fourth power okay again we can unroll this now and ask you know how does this look for a particle at rest again we find the energy is equal to m c square no surprise that's how we started the definitions of this in general we can find that the energy is equal to a relativistic mass times c square which is equal to the rest mass times gamma times z square and that's equal to the rest mass times c square plus k the kinetic energy square all right so this definition i can tell you that this confused me as a student quite a bit this understanding that the mass becomes heavier for particles really one i didn't quite like i just like to think about the fact that the kinetic energy has a relativistic component to the kinetic energy which is owned by the particle in addition to the rest mass of this particle times the square also interesting to note is that for particles at rest particles which are massless like a photon the energy is equal to the momentum times c so if you want to know what the what the energy is of a photon you all you know you need to know what the momentum is of the photon multiplied by c