Physics Lecture Andrew Brandt Monday February 1, 2010 Dr. Andrew Brandt HW1 Assigned due Weds 2/3/10 (you can turn it in on Feb.8, but next HW to be assigned 2/3 will be due 2/10) Relativistic momentum and energy 2/1/2010

Speed of Person Revisited Reference Frames: It’s all relative, 40 mph/backwards? How fast can a person run? Andrew Brandt22/1/2010

Relativistic Momentum So for relativity Galilean transform replaced by Lorentz transform What about Newton’s laws? F=ma= and (classically) Galilean velocity addition ; taking derivative, with v=const gives so But for Lorentz this has extra velocity dependence in denominator, so  accelerations are not equal and a new expression is required for relativistic momentum 3313 Andrew Brandt32/1/2010

Relativistic Momentum Relativistic momentum should have the following properties – should be conserved – as should reduce to classical momentum – what about ? It has these properties – note some texts occasionally define a relativistic mass which increase with velocity, in order to save standard momentum formula In this case the mass becomes infinite as the velocity approaches c ; we will not do this; the mass is always the rest mass in our approach 3313 Andrew Brandt42/1/2010

Relativistic Momentum Example A meteor with mass of 1 kg travels 0.4c What is it made of? Find its momentum; what if it were going twice as fast, what would its momentum be? compare with classical case 3313 Andrew Brandt5 **work old Ex. 1.5 on board ** 2/1/2010

Relativistic Mass and Energy The work done by a constant force over a distance S: For variable force: If object starts from rest and no other forces, work become KE With v=ds/dt Integration by parts with x = v dx = dv and 3313 Andrew Brandt62/1/2010

with 3313 Andrew Brandt7 Relativistic Mass and Energy Use binomial approximation for to show 2/1/2010

Relativistic Energy Ex. 1.6 A stationary high-tech bomb explodes into two fragments each with 1.0 kg mass that move apart at speeds of 0.6 c relative to original bomb. Find the original mass M Andrew Brandt82/1/2010

Relativistic Mass and Energy Mass and energy are related by E=mc 2 Can convert from one to the other One kg of mass = 9x10 16 J of energy (enough to send a 1 million ton payload to the moon!) allvideos.html allvideos.html Is Pluto a planet? Not anymore, it’s a Dwarf planet (part of Kuiper Belt) What about 2003UB313? AKA Eris 3313 Andrew Brandt92/1/2010

More about Energy Energy is conserved, that is, in a given reference frame for an isolated system  E= constant (it may be a different constant in different reference frames) Energy is constant but not invariant, that is, the constant can vary from one frame to another (example an object in its rest frame has less energy than an object in a moving frame). What about mass mc 2 ? Invariant, but not conserved 3313 Andrew Brandt102/1/2010

Relativistic Energy and Momentum Combining and one can obtain These expressions relate the E, p, and m for single particles, but are only valid for a system of particles of mass M if which need not be true Since mc 2 is an invariant quantity, this implies is also an invariant If m=0 this implies E = p = 0, but what if v = c? Then  =1/ 0 and E = p = 0/0, which is undefined! For a massless particle with v = c, then we could have E = pc for example the photon (which unfortunately has the symbol  ) 3313 Andrew Brandt11 and 2/1/2010

Units Use W=qV to define electron volt (eV) a useful energy unit 1 eV = 1.6x C x 1V = 1.6x J Binding energy of hydrogen atom is 13.6 eV Uranium atom releases about 200 MeV when it splits in two (fission) This is not a lot of Joules/fission, but there are a lot of atoms around… Rest mass of proton (m p ) is GeV/c 2 which is a fine unit for mass Units of momentum, p, MeV/c 3313 Andrew Brandt122/1/2010

Relativistic Energy Problems From we can multiply by v to obtain simplifying gives which gives us a useful relation: An electron and a proton are accelerated through 10 MV, find p, v, and  of each (work problem on blackboard) Electron: Proton: Andrew Brandt2/1/2010