Momentum and Energy. Relativity and Astrophysics Lecture 24 Terry Herter. Energy and Momentum Conservation of energy and momentum

Transcription

1 Momentum and Energy Relatiity and Astrohysics Lecture 4 Terry Herter Outline Newtonian Physics Energy and Momentum Conseration of energy and momentum Reading Sacetime Physics: Chater 7 Homework: (due Wed. /04/09) 6-4 and 6-7 (maybe more on Friday) A90-4 Momentum and Energy A90-4

2 Samle Problem 6- (g. 78) Astronaut shouts out Damn! at :00 GMT but after one second (after :00 GMT) a short circuit temorarily disables receier at Mission Control on Earth. Take d = m between Earth and Moon in Earth frame. Does mission control here exletie? No, since c = 30 8 m/sec. Not enough time to trael to Earth before short occurs Could astronauts language hae cause the short? No, since signal can t trael faster than light. How do we classify the sacetime searation? Sacelike sace searation > time searation What is roer distance? s = ( ) (30 8 ) => s =.40 8 m What is shortest distance? The shortest distance is equal to the roer distance A90-4 Momentum and Energy 3 Newtonian Momentum Energy Conseration of Momentum In the absence of a force we hae d F ma m 0 dt So that the momentum () is consered m constant Conseration of Energy Writing the force, F, as the deriatie of a otential, V, we hae dv d dv d F m dr m dr dv md m V E dr dt dr dt The energy, E, of the system is consered. Note: KE m m Conseration laws are ery general in hysics. Symmetries result in a consered quantity Shift symmetry in time => conseration of energy Shift symmetry in sace => conseration of momentum For further info a lace to start is Wikiedia article on conseration of energy The use of conseration laws can greatly simlify the solution of otherwise comlex hysical roblems. A90-4 Momentum and Energy 4 A90-4

3 Conseration of Momentum Examle or m ton m ton m 3 ton 69 mh 60 mh? Suose we two moing objects collide and stick. What is their final elocity? Since momentum is consered we hae m m m m m m 3 3 So that the final elocity is -7 mh. The final, joined mass is moing to the left. Note energy is consered but some of it went into deformation of the object And ossibly heat, sound and light which could carry away momentum, so we assumed this is a small effect. We now look at the case where they bounce off one another. A90-4 Momentum and Energy 5 Conseration of Energy Examle m ton m ton?? 69 mh mh 60 Suose we two moing objects collide elastically (negligible deformation, heating, sounds, etc). What are their final elocities? We use both momentum and energy conseration m m m m m m m m There are two equations with two unknowns. This can be soled by brute force but let s try a somewhat more elegant aroach Define the center of mass (CM) which tracks the net momentum of the system ( = + ) which is consered before and after the collision m m m m which defines and the elocities in CM coordinate system m m,, m m A90-4 Momentum and Energy 6 A90-4 3

4 Energy Examle (cont d) Now define momentum of article relatie to CM mm, m m m Likewise for article,, = ( ) =,. So that, +, = m ton m ton 69 mh 60 mh The conseration equations can now be written as 0,,,, The solution to these equations gies (see right),, The ositie solution is the case where they miss,,, m m m m?? reduced mass mm m m A90-4 Momentum and Energy 7,,,,,, m m m m,,, Energy Examle (cont d) m ton m ton 69 mh 60 mh We can look at the change in momentum (imulse) We can see from definition of, and, that So that,,,,,,,,,,?? And likewise for (switch and in the aboe two equations),, since, =,,, m m m m m m m m,, A90-4 Momentum and Energy 8 A90-4 4

5 Energy Examle (cont d) m ton m ton 69 mh 60 mh Thus we get m m m m m m In the limiting case when m = m we see that?? So that the articles switch elocity when they hae the same mass Such as in ool, the cue ball stos and the imacted ball flies off Note we can use ectors for the elocity and momentum and this all works in and 3 dimensions A90-4 Momentum and Energy 9 Energy Examle (cont d) m ton m ton 69 mh 60 mh We can determine the final elocities for our examle m m m m m m?? 03 mh 6 mh Which shows us that the light object is rebounded at a higher seed (in this case) than it had originally If we consider the case when m >> m then we hae m m m m m m 0 m m m m m >> m A90-4 Momentum and Energy 0 A90-4 5

6 The Power of It All The conseration law all us to sole for quantities without knowing the details We don t hae to know how the objects deform in the sticking case We don t need to know about the details for the collisions at all (for the comletely inelastic and elastic cases) In cases where this a otential we can include this in the energy conseration equations. For instance, using Newton s law of graity we can write a conseration of energy equation which relates elocity to distance from the Earth we don t hae to sole the details of the acceleration For a ertically falling object we hae F GMm r V GMm r m GMm ro r Where the object starts at r o with zero elocity. Note that we lose information. We don t know how long it takes to trael oer this distance just the seed at the end. A90-4 Momentum and Energy A90-4 6