Relativity and Astrophysics Lecture 26 Terry Herter. Reading Spacetime Physics: Chapters 8

Transcription

1 Relativity and Astrophysics Lecture 6 Terry Herter Outline Conservation of Moenergy Particle collision exaple Concept Suary s Collisions Conserved quantities Photons Reading Spacetie Physics: Chapters 8 Reinder: hoework due Wed. /04/09 6-4, 6-7, 7-3, and 7-9 A90-6 A90-6

2 Conservation of oenergy p = -6 8 = 7 8 = 0 p = -5 p = 5 = 3 T = 30 = 0 8 Before p T = 0 (before and after) p = 6 After 8 v 5 /7 v 5/3 v 6 /0 v 6/ 0 Moenergy is conserved in a collision Arrow in spacetie of oenergy has the sae agnitude and direction before and after a collision ach coponent when added together for the two particles is conserved, e.g. T energy & oentu in x (sae for y and z) p T, x p, x p, x p, x p, x A Suary - Moenergy Moenergy is a 4-vector: oenergy Magnitude equals particle and points in the direction of particle worldline Mass is an invariant (identical as viewed fro different inertial fraes). nergy and each coponent of oentu are conserved (preserved before and after a collision) but depend on the inertial frae We have conventional units px py pz or p 4 p c c spacetie displaceent proper tie for that displaceent The space coponents (oentu) are given by: dx p x d rest c dy p y d K dz p z d rest Magnitude: The tie coponent (energy) is given by dt p Note v & d The rest energy, rest, and kinetic energy, K, of a particle are p v v A A90-6

3 The collection of particles (participants) we track in an isolated region of spacetie We will refer to an isolated in which oenergy is conserved Let s look at three siple experients lastic collision Inelastic collision Weighing heat A xperient : lastic Collision v v Before After Consider two identical arbles Marble is oving with speed v while arble is at rest. After a head-on elastic collision, arble is at rest while arble oves off with speed v. Moentu and energy are both conserved Sae total oentu and sae total energy before and after the collision Thus sae total oenergy A A90-6 3

4 xperient : Inelastic Collision v v Before After Consider two identical putty balls Ball and ove towards each other with equal speed v. After a head-on inelastic collision, ball and are both at rest. Moentu is conserved Total oentu before and after the collision is zero, that is, the oentu of the is zero. What about energy? Must also be conserved Before collision: = +K After collision: = rest = M => M syte = + K > Where does extra coe fro? nergy of otion get converted into deforation and heat This increases the of the pair of balls that are stuck together A xperient 3: Weighing Heat If wared and distorted balls of gu have ore than cool and undistorted balls, can we easure the increase? Or can we just heat up and object and weigh the increase This turns out to be a very sall effect xtreely difficult to easure The textbook states that this increase has not been easured haven t found a reference that this has changed. Where is the gained by heat? It is in the relative otion of the particles in the a particle does not gain through otion. Heat is a property Bath water raised fro 5 C to 40 C increases the by part in 0. A A90-6 4

5 Mass of a of Particles The of a is greater when parts of it are oving relative to one another. xaple p v 6 / p 0 Individually p 6 8 v 3/ 5 p / 0 8 v 3/ nergy and oentu are additive. Mass is not additive Mass of the does not agree with the su of the es of its parts Don t ask where the extra (4 units) is located. As where the 0 units of are located. They belong to the as a whole not to any part individually 5 p M M / p 0 0 / 0 5 3/ 5 4 A Mass of is invariant Take the sae but fro a frae oving with particle. The oentu and energy are different as viewed fro new frae 8 v 5 /7 8 v 0 3/ 5 3/ 5 v 3/ 53/ / p p 8 Individually 8 7 /8 / p v 5 7 So the of the is invariant p / p 5 5 / 0 The (total oentu-energy) is the sae in the new frae If the two particles collide then The energy reains the sae (5) The oentu reains the sae (5) The reains the sae (0) M M A A90-6 5

6 a) b) c) d) xaple of es 3 K = 3 K = 5 = 7 = 6 K =kinetic energy, = energy at rest = 6 K = 5 at rest total oentu is not algebraic su of the coponents because they are vectors pointing in directions. p p 4 5 A90-6 a) b) c) d) M rest K p 3 5 / p / / p p 6 35 / 35 / 35 / M 7 8 p / M 899 p p / M 60 A90-6 6