9.3 Collisions

Key Takeaways

welcome back to age 20 special relativity in this section we're going to talk a bit more about collisions i've already seen collisions and study momentum conservation in previous sections so here we can have collisions and we can describe them in the center of mass frame for example where the total momentum is equal to zero so in case of the collision of two particles the momentum of particle one plus the momentum of particle two is equal to zero we have then we can then describe the energy and the momentum of the particles before and after the collision in the lab frame the situation is different here typically we have one particle with some momentum hitting another particle which is at rest but we can also have different types of collisions we describe or characterize elastic collisions where the kinetic energy is conserved and so is the mass so you can think about two billiard balls colliding without any friction in which case you know they don't change their appearance their mass everything is unchanged they just change the direction the total kinetic energy in this collisions are perfectly conserved but we can also have inelastic collisions and there's two different kinds so sticky kinds where the mass after the collision is greater so you have two particles for example maybe they stick together with some you know like play-doh balls um and the kinetic energy after the collision is smaller or you can have explosive collisions where the mass afterwards is smaller maybe you start from one heavy big object and then it's exposed into many smaller ones but the kinetic energy after the collisions is much is smaller those are also conditions so here we want to do an activity and study an inelastic collision so before we have two particles there of billiard balls they're exactly the same and have a velocity u um and after the collision the mass is capital big mass um and we want to describe this collision once in the center of mass frame and one in the laboratory frame and so the question now is uh are the masses and energy cons is energy conserved in those collisions and you're going to describe this in both both reference frames so again stop the video here and try to work this out i already did this so i discussed before i you know in in those collision problems it's always important to really be clear the situation before the collision was a the situation after the condition was b so i'm describing this here first in the center of mass frame where the x i'm just talking about the x component here the x momentum is zero which is equal to the mass times u times gamma minus the mass times u times gamma that's zero the energy before is two times the mass times gamma times c square after the collision the particle is at rest the new one particle is the rest and has an energy large m over at time c square in the laboratory frame the situation is different there is x momentum zero minus m times u prime this is a different velocity times gamma of u prime so here i'm trying to indicate that this gamma is not the same gamma as over here this is a gamma where the velocity is u prime and the energy is the rest mass of the particle at rest plus the mass times gamma times c square of the second particle after the collision the particle has some velocity u and so the momentum in x direction is minus large m times u times gamma of u again and the energy is large m times gamma u times z square okay good so now we can use momentum conservation and find this equation here and from which we can then calculate that the large mass is equal to two times the smaller mass okay so what you find is and this is the relativistic mass you find that as a conclusion that the rest mass is not conserved the mass of this big ball is not simply the mass of the two rest masses or two times the mass of the rest mass you have to consider this gamma factor here it's two times the relativistic mass if you want but you also find that the total energy is conserved in circulation so that the sum of m naught gamma times c square is conserved in deposition irrespective of how you actually in which reference when you discuss the problem i want to close this part of collisions um with a small discussion of units and that will become interesting or important later on when we look at particle physics examples so in particle physics we often talk about units of electron volt in collision experiments or mega electron volts a key electron involves terror electrons so one electron volt is the kinetic energy of a particle with charge e which is accelerated in a potential of one volts so that it corresponds that's the unit of energy and it corresponds to 1.6 times 10 to the minus 19 joule or 1.6 times the minus 90 kilogram meter square over second square the mass of an electron is really really small and those units here are introduced because the mod is mass small and you want to have reasonable numbers to work with so the amount of the electron is 9.11 times 10 to the minus 31 kilogram so if you just rewrite the m naught as equal to m naught c square times one over c square you find that huh now we rewrite this and find that you know the mass is 8 times 10 to the minus 14 joule over c square or in units of electron volts 5 times 10 to the 5 electron volts over c square which is 0.511 mega electron volts over c square or 511 kilo electrons overseas square so when we talk about the mass of an electron we sometimes approach this with natural units in which c square is equal to one and as you know just simply say that the mass of an electron is 511 k kilo electron volts the mass of a muon is um mega electron volts and and so on and so on