Energy and momentum: Collisions and conservation laws

Rest Energy is real: Nuclear fission Initial mass of a neutron and a 235 U nucleus. Final products have less mass, but much more kinetic energy. Conversion of mass to kinetic energy. Oh yes, and more neutrons, so the reaction can run wild (chain reaction!). Protons: U=92, Ba = 56, Kr=36

Suppose that isolated object A has rest mass = 9m 0 and speed v A =0.8c (  A =5/3). Object B has mass 12m 0 and speed v B =−0.6c (  B =5/4). The objects collide and stick together (completely inelastic collision) From Physics 1110 we know all collisions conserve momentum A relativistic inelastic collision How about energy conservation? A)The above collision will conserve total kinetic energy B)The above collision will conserve total rest energy E rest =m A c 2 + m B c 2 C)The above collision will conserve total energy E total =  A m A c 2 +  B m B c 2 D)It is an inelastic collision so heat will be generated. None of the above will hold.

Classically, what is the total initial momentum? Let’s start with momentum conservation. What is the total relativistic momentum? So it does not end at rest as predicted classically! Suppose that object A has rest mass = 9m 0 and speed v A =0.8c (  A =5/3). Object B has mass 12m 0 and speed v B =−0.6c (  B =5/4). A relativistic inelastic collision Momentum conservation gives us: Remember that m f may not be m A +m B as it would be classically.

Now let’s look at the total energy. The initial energy is So conservation of energy gives us: Dividing these two equations: or so Furthermore: conservation of energy equation for m f : so we can solve the Object A has rest mass = 9m 0 and speed v A =0.8c (  A =5/3). Object B has mass 12m 0 and speed v B =−0.6c (  B =5/4). A relativistic inelastic collision

Classically, total momentum is 0 but in reality it is Classically, but in reality, so 8.85m 0 of mass is gained! So the change in KE is The initial and final kinetic energies are: The “lost” kinetic energy appears as gained mass in the total energy A relativistic inelastic collision

Really, mass gets created! CERN in Geneva, Switzerland Before the LHC (Large Hadron Collider) CERN operated LEP, the Large Electron-Positron collider in the same underground tunnel. Electron and positrons have a mass of 9x kg. They were accelerated to very high energies so when they annihilate, they create a Z 0 particle with a mass of 1.6x kg.

8 A proton has a mass of 938 MeV/c 2. What is this in kg? An important unit of energy is the electron-volt (eV). It’s the energy obtained by an electron moving through 1 V. It is not an SI unit but is very common. ΔE = qΔV = 1 eV = C 1 V = J Also use eV/c or MeV/c units for momentum Since mc 2 is a unit of energy, dividing energy by c 2 gives a unit of mass. Also, dividing energy by c gives a unit of momentum.

At what speed is the total energy of a particle equal to twice its rest mass energy? A. 0 B. 0.7c C. 0.87c D. 0.94c E. c To have total energy equal to twice the rest mass energy, need  =2 Solve for . so you need to be moving pretty fast to get your kinetic energy close to your rest mass energy!