Relativistic MomentumRelativistic ForceRelativistic EnergyConservation of Energy:Relationship between Energy and MomentumThe Electron-Volt Energy UnitInvariant MassBinding EnergyReaction EnergyPHY2061 Enriched Physics 2 Lecture Notes Relativity 4 Relativity 4 Disclaimer: These lecture notes are not meant to replace the course textbook. The content may be incomplete. Some topics may be unclear. These notes are only meant to be a study aid and a supplement to your own notes. Please report any inaccuracies to the professor. Relativistic Momentum Newton’s 2nd Law can be written in the form Fp=ddt where the non-relativistic momentum of a body is pu=m where ux=ddt. However, because of the Lorentz transformation equations, ddtx is measured differently in different inertial frames. Thus, Newton’s 2nd Law would not have the same form in different frames. We need a new definition of momentum to retain the definition of force as a change in momentum. Suppose px= mddτ, where τ is the proper time in the object’s rest frame. Every observer will agree on which frame is the rest frame. Also, since ′=′=yyzzand , the transverse momentum (py and pz) will be invariant for a Lorentz transformation along the x axis. (This would not be the case if we did not use the proper time in the definition). We can rewrite this momentum definition as follows: Recall that momentum is a vector quantity. Conservation of momentum, which still applies in Special Relativity, implies that each component of momentum is conserved.pxx===⇒ =mddmddtdtdtdtdττγττγ From time dilation 221 1/uumucγγ==−puNote that u is the velocity of the object in a reference frame, not the velocity of a reference frame relative to another. In this definition of momentum, the mass m=m0 is the “rest mass”. That is, it is the mass of an object in its rest frame. Sometimes γm is referred to as the “relativistic mass”, such that we can retain the Newtonian definition of momentum as pu=m. In this sense, the mass of an object grows as its velocity increases. But this convenient trick can be problematic. As we shall see, the kinetic energy, for example, is not ½ mv2. D. Acosta Page 1 10/11/2005PHY2061 Enriched Physics 2 Lecture Notes Relativity 4 Relativistic Force With the previous relativistic definition for momentum, we can retain the usual definition for force: ()221 where = and 1/uuddd d dmmdt dt d dt dtucγγτ⎛⎞== = =⎜⎟⎝⎠−px xFuu It is useful to consider how force transforms under a Lorentz Transformation: x y z Sx' y' z'S' v u⊥′ According to the addition of velocity formulae, the transformation of the velocity perpendicular to the direction of the Lorentz Transformation is: ()2221 where 1/1/vvxvcvu cγγ⊥⊥′==′−+uu So for the perpendicular force, which can be written as: dd dmmdt d dττ⊥⊥==xFu it transforms as: ()()()22211/ 1/1/vx vxvxddmmddvu c vu cvu cττγγγ⊥⊥⊥⊥⊥′′==′′++′=′+uuFFF where we assume no acceleration in the direction parallel to the transformation D. Acosta Page 2 10/11/2005PHY2061 Enriched Physics 2 Lecture Notes Relativity 4 Relativistic Energy Now work is defined as force applied over a distance. It corresponds to the expended energy to accelerate a body. If the force and path are constant, WFd=⋅ More generally, if the force and path vary, then a line integral must be performed from initial position 1 to final position 2. Wd1212=⋅z Fs The work applied to a body translates to a change in the kinetic energy since energy must be conserved. If we assume that the body is initially at rest, then the final kinetic energy is equal to the work expended: d s u 1 2 WKddtmdt dKmdtddtKm uduKmUmudumU muduucmU mc u cmU mc U c mcmU mc U c mcUUUU== ⋅ ==⋅==−=−−=+−=+− −=+− −zzzzzγγγγγγγγγγ where we have used Integrate by parts: uu s uuuafafafch020222022 22022 22222 22 21111////dt You can check this integral by differentiation Thus, we get for the relativistic kinetic energy: Kmcmc mc=−=−γγ 221af2 This final expression for the kinetic energy looks like nothing like the non-relativistic equation K. However, if we consider velocities much less than the speed of light, we can see the correspondence: mu=122D. Acosta Page 3 10/11/2005PHY2061 Enriched Physics 2 Lecture Notes Relativity 4 γγ=− ≈+ +⇒=− ≈ <<−11121121222122222222uc ucKmcucmc mu u c///chaf" using the binomial expansion= for So at low velocities there is no difference between the definition of kinetic energy in Special Relativity from that in Newtonian Mechanics. Now let’s consider the opposite limit when the velocity approaches the speed of light. In that case, the kinetic energy becomes infinite as the relativistic factor γ goes to infinity. This is another way of saying that objects cannot exceed the speed of light, because it would take an infinite amount of energy. Now let’s rewrite the equation involving the kinetic energy: EmcKmc≡=+γ22This equation has the form of kinetic energy plus potential energy equals total energy. What is the potential energy? It is the term: Emc02= which we refer to as the rest energy. As you know, this is Einstein’s famous equation that tells us that mass is another form of energy. Mass can be converted into energy and vice versa. How much energy? Let’s see: Example: Suppose that a 1 kg mass moves at a velocity u = 1 m/s. The kinetic energy is ½ m u2 = ½ J. (We can use the non-relativistic equation because the velocity is much much smaller than the speed of light.) The rest mass energy is mc2190 106=×. J. Clearly there is a tremendous amount of energy in 1 kg of mass. That is why nuclear weapons have the power that they do, because they convert a significant amount of mass into energy. Conservation of Energy: We have learned in earlier physics courses that kinetic energy does not have to be conserved in an inelastic collision. Likewise, mass does not have to be conserved since it can be converted into energy. However, the total energy (kinetic, rest mass, and all other potential energy forms) is always conserved in Special Relativity. Momentum and energy are conserved for both elastic and inelastic collisions when the relativistic definitions are used. D. Acosta Page 4 10/11/2005PHY2061 Enriched Physics 2 Lecture Notes Relativity 4
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