2136:
2669:
1423:
38:
1861:
2402:
1156:
676:
2131:{\displaystyle {\begin{aligned}\mathbf {p} _{1}^{\prime }&=m_{1}\mathbf {u} _{1}^{\prime }\\&=m_{1}\left(\mathbf {u} _{1}-\mathbf {V} \right)={\frac {m_{1}m_{2}}{m_{1}+m_{2}}}\left(\mathbf {u} _{1}-\mathbf {u} _{2}\right)\\&=-m_{2}\mathbf {u} _{2}^{\prime }=-\mathbf {p} _{2}^{\prime }\\\end{aligned}}}
2367:
This is a substantially simpler calculation of the momenta of both particles; the reduced mass and relative velocity can be calculated from the initial velocities in the lab frame and the masses, and the momentum of one particle is simply the negative of the other. The calculation can be repeated for
1854:
returns the lab frame equation above, demonstrating any frame (including the COM frame) may be used to calculate the momenta of the particles. It has been established that the velocity of the COM frame can be removed from the calculation using the above frame, so the momenta of the particles in the
2664:{\displaystyle {\begin{aligned}{\frac {{\rm {d}}\mathbf {R} }{{\rm {d}}t}}&={\frac {\rm {d}}{{\rm {d}}t}}\left({\frac {m_{1}\mathbf {r} _{1}+m_{2}\mathbf {r} _{2}}{m_{1}+m_{2}}}\right)\\&={\frac {m_{1}\mathbf {v} _{1}+m_{2}\mathbf {v} _{2}}{m_{1}+m_{2}}}\\&=\mathbf {V} \\\end{aligned}}}
1418:{\displaystyle {\begin{aligned}{\frac {{\rm {d}}\mathbf {R} }{{\rm {d}}t}}&={\frac {\rm {d}}{{\rm {d}}t}}\left({\frac {m_{1}\mathbf {r} _{1}+m_{2}\mathbf {r} _{2}}{m_{1}+m_{2}}}\right)\\&={\frac {m_{1}\mathbf {u} _{1}+m_{2}\mathbf {u} _{2}}{m_{1}+m_{2}}}\\&=\mathbf {V} \\\end{aligned}}}
1703:
3227:
1845:
3083:
1133:
367:
2922:
3318:
2765:
1519:
905:
2362:
3397:
1544:
3106:
1728:
2937:
2201:
356:
1037:
671:{\displaystyle \sum _{i}p'_{i}=\sum _{i}m_{i}v'_{i}=\sum _{i}m_{i}(v_{i}-V_{c})=\sum _{i}m_{i}v_{i}-\sum _{i}m_{i}{\frac {\sum _{j}m_{j}v_{j}}{\sum _{j}m_{j}}}=\sum _{i}m_{i}v_{i}-\sum _{j}m_{j}v_{j}=0.}
944:
system must travel at the speed of light in any frame, and always possesses a net momentum. Its energy is—for each reference frame—equal to the magnitude of momentum multiplied by the speed of light:
2776:
2285:
3111:
2407:
1866:
1161:
1031:
respectively. The transformations are applied to take the velocity of the frame from the velocity of each particle from the lab frame (unprimed quantities) to the COM frame (primed quantities):
3238:
793:
262:
2688:
1442:
808:
2296:
977:
3333:
162:
of a system is not a location, but a collection of relative momenta/velocities: a reference frame. Thus "center of momentum" is a short for "center-of-momentum
1698:{\displaystyle \mathbf {V} ={\frac {\mathbf {p} _{1}+\mathbf {p} _{2}}{m_{1}+m_{2}}}={\frac {m_{1}\mathbf {u} _{1}+m_{2}\mathbf {u} _{2}}{m_{1}+m_{2}}}}
3222:{\displaystyle {\begin{aligned}\mathbf {P} &=\mathbf {p} _{1}+\mathbf {p} _{2}\\&=(m_{1}+m_{2})\mathbf {V} \\&=M\mathbf {V} \end{aligned}}}
1840:{\displaystyle \mathbf {p} _{1}^{\prime }+\mathbf {p} _{2}^{\prime }=m_{1}\mathbf {u} _{1}^{\prime }+m_{2}\mathbf {u} _{2}^{\prime }={\boldsymbol {0}}}
3078:{\displaystyle m_{1}\mathbf {u} _{1}=m_{1}\mathbf {v} _{1}=m_{1}\mathbf {V} ,\quad m_{2}\mathbf {u} _{2}=m_{2}\mathbf {v} _{2}=m_{2}\mathbf {V} }
2151:
706:
273:
198:
The center of momentum frame is defined as the inertial frame in which the sum of the linear momenta of all particles is equal to 0. Let
1128:{\displaystyle \mathbf {u} _{1}^{\prime }=\mathbf {u} _{1}-\mathbf {V} ,\quad \mathbf {u} _{2}^{\prime }=\mathbf {u} _{2}-\mathbf {V} }
102:
74:
55:
2917:{\displaystyle m_{1}\mathbf {u} _{1}+m_{2}\mathbf {u} _{2}=m_{1}\mathbf {v} _{1}+m_{2}\mathbf {v} _{2}=(m_{1}+m_{2})\mathbf {V} }
81:
3475:
3440:
2216:
88:
936:) do not have COM frames, because there is no frame in which they have zero net momentum. Due to the invariance of the
17:
3313:{\displaystyle \mathbf {p} _{1}^{\prime }=-\mathbf {p} _{2}^{\prime }=\mu \Delta \mathbf {v} =\mu \Delta \mathbf {u} }
3456:
121:
70:
741:
987:
An example of the usage of this frame is given below – in a two-body collision, not necessarily elastic (where
59:
3408:
177:(which is a single point) remains at the origin. In all center-of-momentum frames, the center of mass is at
3503:
3493:
1524:
The same results can be obtained by applying momentum conservation in the lab frame, where the momenta are
3498:
1855:
COM frame can be expressed in terms of the quantities in the lab frame (i.e. the given initial values):
361:
is the velocity of the mass center. The total momentum in the center-of-momentum system then vanishes:
3508:
95:
991:
is conserved). The COM frame can be used to find the momentum of the particles much easier than in a
1000:
996:
690:
2760:{\displaystyle m_{1}\mathbf {v} _{1}^{\prime }+m_{2}\mathbf {v} _{2}^{\prime }={\boldsymbol {0}}}
1708:
and in the COM frame, where it is asserted definitively that the total momenta of the particles,
1514:{\displaystyle m_{1}\mathbf {u} _{1}^{\prime }+m_{2}\mathbf {u} _{2}^{\prime }={\boldsymbol {0}}}
220:
207:
48:
900:{\displaystyle m_{0}{}^{2}=\left({\frac {E}{c^{2}}}\right)^{2}-\left({\frac {p}{c}}\right)^{2},}
2357:{\displaystyle \mathbf {p} _{1}^{\prime }=-\mathbf {p} _{2}^{\prime }=\mu \Delta \mathbf {u} }
950:
933:
3392:{\displaystyle \Delta \mathbf {v} =\mathbf {v} _{1}-\mathbf {v} _{2}=\Delta \mathbf {u} .}
995:: the frame where the measurement or calculation is done. The situation is analyzed using
8:
3513:
702:
182:
155:
3518:
3471:
3452:
3436:
3324:
2142:
941:
802:
of the system is given in any inertial frame by the relativistic invariant relation
1003:(for generality, rather than kinetic energies alone), for two particles of mass
705:, the COM frame exists for an isolated massive system. This is a consequence of
937:
799:
732:
724:
178:
174:
151:
29:
Unique inertial frame in which the total momentum of a physical system vanishes
2396:, since after the collision the velocities still satisfy the above equations:
3487:
2207:
3413:
1146:
is the velocity of the COM, i.e. the time derivative of the COM location
711:
158:
of the system vanishes. It is unique up to velocity, but not origin. The
185:, the COM frame is necessarily unique only when the system is isolated.
930:
181:, but it is not necessarily at the origin of the coordinate system. In
2196:{\displaystyle \Delta \mathbf {u} =\mathbf {u} _{1}-\mathbf {u} _{2}}
992:
922:
918:) vanishes and thus the total energy coincides with the rest energy.
728:
37:
3451:
Classical
Mechanics, T.W.B. Kibble, European Physics Series, 1973,
351:{\displaystyle V_{c}={\frac {\sum _{i}m_{i}v_{i}}{\sum _{i}m_{i}}}}
926:
135:
3470:, D. Kleppner, R.J. Kolenkow, Cambridge University Press, 2010,
3435:
Dynamics and
Relativity, J.R. Forshaw, A.G. Smith, Wiley, 2009,
682:
2770:
In the lab frame, the conservation of momentum fully reads:
709:. In the COM frame the total energy of the system is the
206:′ denote the center-of-momentum reference frame. Using a
2280:{\displaystyle \mu ={\frac {m_{1}m_{2}}{m_{1}+m_{2}}}}
169:
A special case of the center-of-momentum frame is the
3336:
3241:
3109:
2940:
2779:
2691:
2405:
2299:
2219:
2154:
1864:
1731:
1547:
1445:
1159:
1040:
953:
811:
744:
370:
276:
223:
2290:
so the momenta of the particles compactly reduce to
3431:
3429:
62:. Unsourced material may be challenged and removed.
3391:
3312:
3221:
3077:
2916:
2759:
2663:
2356:
2279:
2195:
2130:
1839:
1697:
1513:
1417:
1127:
1017:, moving at initial velocities (before collision)
971:
899:
787:
670:
350:
256:
3092:multiplied by the velocity of the centre of mass
3485:
3426:
1150:(position of the center of mass of the system):
929:moving in a single direction, or, equivalently,
715:, and this quantity (when divided by the factor
788:{\displaystyle m_{0}={\frac {E_{0}}{c^{2}}}.}
3088:instead, it simply indicates the total mass
202:denote the laboratory reference system and
1850:Using the COM frame equation to solve for
921:Systems that have nonzero energy but zero
910:but for zero momentum the momentum term (
122:Learn how and when to remove this message
1142:is the velocity of the COM frame. Since
3327:in the lab frame of particle 1 to 2 is
2145:in the lab frame of particle 1 to 2 is
14:
3486:
3232:Similar analysis to the above obtains
696:
60:adding citations to reliable sources
31:
2682:, this implies after the collision
2674:so at the origin of the COM frame,
2382:in place of the initial velocities
1428:so at the origin of the COM frame,
982:
24:
3378:
3337:
3302:
3288:
3277:
3254:
2744:
2714:
2457:
2450:
2429:
2415:
2346:
2335:
2312:
2155:
2119:
2096:
1915:
1881:
1824:
1794:
1764:
1744:
1498:
1468:
1211:
1204:
1183:
1169:
1097:
1053:
25:
3530:
173:: an inertial frame in which the
3382:
3365:
3350:
3341:
3306:
3292:
3267:
3244:
3211:
3193:
3143:
3128:
3115:
3071:
3047:
3022:
3002:
2978:
2953:
2910:
2867:
2842:
2817:
2792:
2753:
2734:
2704:
2653:
2604:
2579:
2512:
2487:
2421:
2350:
2325:
2302:
2183:
2168:
2159:
2109:
2086:
2046:
2031:
1961:
1947:
1905:
1871:
1833:
1814:
1784:
1754:
1734:
1657:
1632:
1576:
1561:
1549:
1507:
1488:
1458:
1407:
1358:
1333:
1266:
1241:
1175:
1121:
1107:
1087:
1077:
1063:
1043:
36:
3009:
1084:
47:needs additional citations for
3461:
3445:
3189:
3163:
2906:
2880:
479:
453:
13:
1:
3419:
3409:Laboratory frame of reference
188:
3468:An Introduction to Mechanics
7:
3402:
257:{\displaystyle v'=v-V_{c},}
210:, the particle velocity in
10:
3535:
193:
71:"Center-of-momentum frame"
691:inertial reference frames
1001:conservation of momentum
997:Galilean transformations
140:center-of-momentum frame
208:Galilean transformation
3393:
3314:
3223:
3096:is the total momentum
3079:
2918:
2761:
2665:
2358:
2281:
2197:
2132:
1841:
1699:
1515:
1419:
1129:
973:
901:
789:
672:
352:
258:
3394:
3315:
3224:
3080:
2919:
2762:
2666:
2359:
2282:
2198:
2133:
1842:
1700:
1516:
1420:
1130:
974:
972:{\displaystyle E=pc.}
934:electromagnetic waves
902:
790:
685:of the system is the
673:
353:
259:
3334:
3239:
3107:
2938:
2777:
2689:
2403:
2297:
2217:
2152:
1862:
1729:
1545:
1443:
1157:
1038:
951:
809:
742:
368:
274:
221:
171:center-of-mass frame
56:improve this article
3504:Frames of reference
3494:Classical mechanics
3281:
3258:
2927:This equation does
2748:
2718:
2339:
2316:
2123:
2100:
1919:
1885:
1828:
1798:
1768:
1748:
1502:
1472:
1101:
1057:
429:
393:
154:in which the total
148:zero-momentum frame
3499:Coordinate systems
3389:
3310:
3265:
3242:
3219:
3217:
3075:
2914:
2757:
2732:
2702:
2661:
2659:
2354:
2323:
2300:
2277:
2193:
2128:
2126:
2107:
2084:
1903:
1869:
1837:
1812:
1782:
1752:
1732:
1695:
1511:
1486:
1456:
1415:
1413:
1125:
1085:
1041:
969:
897:
785:
697:Special relativity
668:
641:
608:
582:
550:
527:
494:
442:
417:
406:
381:
380:
348:
334:
302:
254:
183:special relativity
160:center of momentum
18:Center of momentum
3509:Geometric centers
3476:978-0-521-19821-9
3441:978-0-470-01460-8
3325:relative velocity
2640:
2548:
2466:
2438:
2368:final velocities
2275:
2143:relative velocity
2022:
1693:
1612:
1394:
1302:
1220:
1192:
882:
854:
780:
735:) of the system:
707:Noether's theorem
689:as seen from all
632:
599:
594:
573:
541:
518:
485:
433:
397:
371:
346:
325:
293:
146:), also known as
132:
131:
124:
106:
16:(Redirected from
3526:
3478:
3465:
3459:
3449:
3443:
3433:
3398:
3396:
3395:
3390:
3385:
3374:
3373:
3368:
3359:
3358:
3353:
3344:
3323:where the final
3319:
3317:
3316:
3311:
3309:
3295:
3280:
3275:
3270:
3257:
3252:
3247:
3228:
3226:
3225:
3220:
3218:
3214:
3200:
3196:
3188:
3187:
3175:
3174:
3156:
3152:
3151:
3146:
3137:
3136:
3131:
3118:
3084:
3082:
3081:
3076:
3074:
3069:
3068:
3056:
3055:
3050:
3044:
3043:
3031:
3030:
3025:
3019:
3018:
3005:
3000:
2999:
2987:
2986:
2981:
2975:
2974:
2962:
2961:
2956:
2950:
2949:
2923:
2921:
2920:
2915:
2913:
2905:
2904:
2892:
2891:
2876:
2875:
2870:
2864:
2863:
2851:
2850:
2845:
2839:
2838:
2826:
2825:
2820:
2814:
2813:
2801:
2800:
2795:
2789:
2788:
2766:
2764:
2763:
2758:
2756:
2747:
2742:
2737:
2731:
2730:
2717:
2712:
2707:
2701:
2700:
2670:
2668:
2667:
2662:
2660:
2656:
2645:
2641:
2639:
2638:
2637:
2625:
2624:
2614:
2613:
2612:
2607:
2601:
2600:
2588:
2587:
2582:
2576:
2575:
2565:
2557:
2553:
2549:
2547:
2546:
2545:
2533:
2532:
2522:
2521:
2520:
2515:
2509:
2508:
2496:
2495:
2490:
2484:
2483:
2473:
2467:
2465:
2461:
2460:
2453:
2448:
2439:
2437:
2433:
2432:
2425:
2424:
2419:
2418:
2411:
2363:
2361:
2360:
2355:
2353:
2338:
2333:
2328:
2315:
2310:
2305:
2286:
2284:
2283:
2278:
2276:
2274:
2273:
2272:
2260:
2259:
2249:
2248:
2247:
2238:
2237:
2227:
2202:
2200:
2199:
2194:
2192:
2191:
2186:
2177:
2176:
2171:
2162:
2137:
2135:
2134:
2129:
2127:
2122:
2117:
2112:
2099:
2094:
2089:
2083:
2082:
2064:
2060:
2056:
2055:
2054:
2049:
2040:
2039:
2034:
2023:
2021:
2020:
2019:
2007:
2006:
1996:
1995:
1994:
1985:
1984:
1974:
1969:
1965:
1964:
1956:
1955:
1950:
1939:
1938:
1923:
1918:
1913:
1908:
1902:
1901:
1884:
1879:
1874:
1846:
1844:
1843:
1838:
1836:
1827:
1822:
1817:
1811:
1810:
1797:
1792:
1787:
1781:
1780:
1767:
1762:
1757:
1747:
1742:
1737:
1704:
1702:
1701:
1696:
1694:
1692:
1691:
1690:
1678:
1677:
1667:
1666:
1665:
1660:
1654:
1653:
1641:
1640:
1635:
1629:
1628:
1618:
1613:
1611:
1610:
1609:
1597:
1596:
1586:
1585:
1584:
1579:
1570:
1569:
1564:
1557:
1552:
1520:
1518:
1517:
1512:
1510:
1501:
1496:
1491:
1485:
1484:
1471:
1466:
1461:
1455:
1454:
1424:
1422:
1421:
1416:
1414:
1410:
1399:
1395:
1393:
1392:
1391:
1379:
1378:
1368:
1367:
1366:
1361:
1355:
1354:
1342:
1341:
1336:
1330:
1329:
1319:
1311:
1307:
1303:
1301:
1300:
1299:
1287:
1286:
1276:
1275:
1274:
1269:
1263:
1262:
1250:
1249:
1244:
1238:
1237:
1227:
1221:
1219:
1215:
1214:
1207:
1202:
1193:
1191:
1187:
1186:
1179:
1178:
1173:
1172:
1165:
1134:
1132:
1131:
1126:
1124:
1116:
1115:
1110:
1100:
1095:
1090:
1080:
1072:
1071:
1066:
1056:
1051:
1046:
983:Two-body problem
978:
976:
975:
970:
906:
904:
903:
898:
893:
892:
887:
883:
875:
865:
864:
859:
855:
853:
852:
840:
830:
829:
824:
821:
820:
794:
792:
791:
786:
781:
779:
778:
769:
768:
759:
754:
753:
681:Also, the total
677:
675:
674:
669:
661:
660:
651:
650:
640:
628:
627:
618:
617:
607:
595:
593:
592:
591:
581:
571:
570:
569:
560:
559:
549:
539:
537:
536:
526:
514:
513:
504:
503:
493:
478:
477:
465:
464:
452:
451:
441:
425:
416:
415:
405:
389:
379:
357:
355:
354:
349:
347:
345:
344:
343:
333:
323:
322:
321:
312:
311:
301:
291:
286:
285:
263:
261:
260:
255:
250:
249:
231:
127:
120:
116:
113:
107:
105:
64:
40:
32:
21:
3534:
3533:
3529:
3528:
3527:
3525:
3524:
3523:
3484:
3483:
3482:
3481:
3466:
3462:
3450:
3446:
3434:
3427:
3422:
3405:
3381:
3369:
3364:
3363:
3354:
3349:
3348:
3340:
3335:
3332:
3331:
3305:
3291:
3276:
3271:
3266:
3253:
3248:
3243:
3240:
3237:
3236:
3216:
3215:
3210:
3198:
3197:
3192:
3183:
3179:
3170:
3166:
3154:
3153:
3147:
3142:
3141:
3132:
3127:
3126:
3119:
3114:
3110:
3108:
3105:
3104:
3100:of the system:
3070:
3064:
3060:
3051:
3046:
3045:
3039:
3035:
3026:
3021:
3020:
3014:
3010:
3001:
2995:
2991:
2982:
2977:
2976:
2970:
2966:
2957:
2952:
2951:
2945:
2941:
2939:
2936:
2935:
2909:
2900:
2896:
2887:
2883:
2871:
2866:
2865:
2859:
2855:
2846:
2841:
2840:
2834:
2830:
2821:
2816:
2815:
2809:
2805:
2796:
2791:
2790:
2784:
2780:
2778:
2775:
2774:
2752:
2743:
2738:
2733:
2726:
2722:
2713:
2708:
2703:
2696:
2692:
2690:
2687:
2686:
2658:
2657:
2652:
2643:
2642:
2633:
2629:
2620:
2616:
2615:
2608:
2603:
2602:
2596:
2592:
2583:
2578:
2577:
2571:
2567:
2566:
2564:
2555:
2554:
2541:
2537:
2528:
2524:
2523:
2516:
2511:
2510:
2504:
2500:
2491:
2486:
2485:
2479:
2475:
2474:
2472:
2468:
2456:
2455:
2454:
2449:
2447:
2440:
2428:
2427:
2426:
2420:
2414:
2413:
2412:
2410:
2406:
2404:
2401:
2400:
2395:
2388:
2381:
2374:
2349:
2334:
2329:
2324:
2311:
2306:
2301:
2298:
2295:
2294:
2268:
2264:
2255:
2251:
2250:
2243:
2239:
2233:
2229:
2228:
2226:
2218:
2215:
2214:
2206:and the 2-body
2187:
2182:
2181:
2172:
2167:
2166:
2158:
2153:
2150:
2149:
2125:
2124:
2118:
2113:
2108:
2095:
2090:
2085:
2078:
2074:
2062:
2061:
2050:
2045:
2044:
2035:
2030:
2029:
2028:
2024:
2015:
2011:
2002:
1998:
1997:
1990:
1986:
1980:
1976:
1975:
1973:
1960:
1951:
1946:
1945:
1944:
1940:
1934:
1930:
1921:
1920:
1914:
1909:
1904:
1897:
1893:
1886:
1880:
1875:
1870:
1865:
1863:
1860:
1859:
1832:
1823:
1818:
1813:
1806:
1802:
1793:
1788:
1783:
1776:
1772:
1763:
1758:
1753:
1743:
1738:
1733:
1730:
1727:
1726:
1721:
1714:
1686:
1682:
1673:
1669:
1668:
1661:
1656:
1655:
1649:
1645:
1636:
1631:
1630:
1624:
1620:
1619:
1617:
1605:
1601:
1592:
1588:
1587:
1580:
1575:
1574:
1565:
1560:
1559:
1558:
1556:
1548:
1546:
1543:
1542:
1537:
1530:
1506:
1497:
1492:
1487:
1480:
1476:
1467:
1462:
1457:
1450:
1446:
1444:
1441:
1440:
1436:, this implies
1412:
1411:
1406:
1397:
1396:
1387:
1383:
1374:
1370:
1369:
1362:
1357:
1356:
1350:
1346:
1337:
1332:
1331:
1325:
1321:
1320:
1318:
1309:
1308:
1295:
1291:
1282:
1278:
1277:
1270:
1265:
1264:
1258:
1254:
1245:
1240:
1239:
1233:
1229:
1228:
1226:
1222:
1210:
1209:
1208:
1203:
1201:
1194:
1182:
1181:
1180:
1174:
1168:
1167:
1166:
1164:
1160:
1158:
1155:
1154:
1120:
1111:
1106:
1105:
1096:
1091:
1086:
1076:
1067:
1062:
1061:
1052:
1047:
1042:
1039:
1036:
1035:
1030:
1023:
1016:
1009:
985:
952:
949:
948:
888:
874:
870:
869:
860:
848:
844:
839:
835:
834:
825:
823:
822:
816:
812:
810:
807:
806:
774:
770:
764:
760:
758:
749:
745:
743:
740:
739:
699:
656:
652:
646:
642:
636:
623:
619:
613:
609:
603:
587:
583:
577:
572:
565:
561:
555:
551:
545:
540:
538:
532:
528:
522:
509:
505:
499:
495:
489:
473:
469:
460:
456:
447:
443:
437:
421:
411:
407:
401:
385:
375:
369:
366:
365:
339:
335:
329:
324:
317:
313:
307:
303:
297:
292:
290:
281:
277:
275:
272:
271:
245:
241:
224:
222:
219:
218:
196:
191:
128:
117:
111:
108:
65:
63:
53:
41:
30:
23:
22:
15:
12:
11:
5:
3532:
3522:
3521:
3516:
3511:
3506:
3501:
3496:
3480:
3479:
3460:
3444:
3424:
3423:
3421:
3418:
3417:
3416:
3411:
3404:
3401:
3400:
3399:
3388:
3384:
3380:
3377:
3372:
3367:
3362:
3357:
3352:
3347:
3343:
3339:
3321:
3320:
3308:
3304:
3301:
3298:
3294:
3290:
3287:
3284:
3279:
3274:
3269:
3264:
3261:
3256:
3251:
3246:
3230:
3229:
3213:
3209:
3206:
3203:
3201:
3199:
3195:
3191:
3186:
3182:
3178:
3173:
3169:
3165:
3162:
3159:
3157:
3155:
3150:
3145:
3140:
3135:
3130:
3125:
3122:
3120:
3117:
3113:
3112:
3086:
3085:
3073:
3067:
3063:
3059:
3054:
3049:
3042:
3038:
3034:
3029:
3024:
3017:
3013:
3008:
3004:
2998:
2994:
2990:
2985:
2980:
2973:
2969:
2965:
2960:
2955:
2948:
2944:
2925:
2924:
2912:
2908:
2903:
2899:
2895:
2890:
2886:
2882:
2879:
2874:
2869:
2862:
2858:
2854:
2849:
2844:
2837:
2833:
2829:
2824:
2819:
2812:
2808:
2804:
2799:
2794:
2787:
2783:
2768:
2767:
2755:
2751:
2746:
2741:
2736:
2729:
2725:
2721:
2716:
2711:
2706:
2699:
2695:
2672:
2671:
2655:
2651:
2648:
2646:
2644:
2636:
2632:
2628:
2623:
2619:
2611:
2606:
2599:
2595:
2591:
2586:
2581:
2574:
2570:
2563:
2560:
2558:
2556:
2552:
2544:
2540:
2536:
2531:
2527:
2519:
2514:
2507:
2503:
2499:
2494:
2489:
2482:
2478:
2471:
2464:
2459:
2452:
2446:
2443:
2441:
2436:
2431:
2423:
2417:
2409:
2408:
2393:
2386:
2379:
2372:
2365:
2364:
2352:
2348:
2345:
2342:
2337:
2332:
2327:
2322:
2319:
2314:
2309:
2304:
2288:
2287:
2271:
2267:
2263:
2258:
2254:
2246:
2242:
2236:
2232:
2225:
2222:
2204:
2203:
2190:
2185:
2180:
2175:
2170:
2165:
2161:
2157:
2139:
2138:
2121:
2116:
2111:
2106:
2103:
2098:
2093:
2088:
2081:
2077:
2073:
2070:
2067:
2065:
2063:
2059:
2053:
2048:
2043:
2038:
2033:
2027:
2018:
2014:
2010:
2005:
2001:
1993:
1989:
1983:
1979:
1972:
1968:
1963:
1959:
1954:
1949:
1943:
1937:
1933:
1929:
1926:
1924:
1922:
1917:
1912:
1907:
1900:
1896:
1892:
1889:
1887:
1883:
1878:
1873:
1868:
1867:
1848:
1847:
1835:
1831:
1826:
1821:
1816:
1809:
1805:
1801:
1796:
1791:
1786:
1779:
1775:
1771:
1766:
1761:
1756:
1751:
1746:
1741:
1736:
1719:
1712:
1706:
1705:
1689:
1685:
1681:
1676:
1672:
1664:
1659:
1652:
1648:
1644:
1639:
1634:
1627:
1623:
1616:
1608:
1604:
1600:
1595:
1591:
1583:
1578:
1573:
1568:
1563:
1555:
1551:
1535:
1528:
1522:
1521:
1509:
1505:
1500:
1495:
1490:
1483:
1479:
1475:
1470:
1465:
1460:
1453:
1449:
1426:
1425:
1409:
1405:
1402:
1400:
1398:
1390:
1386:
1382:
1377:
1373:
1365:
1360:
1353:
1349:
1345:
1340:
1335:
1328:
1324:
1317:
1314:
1312:
1310:
1306:
1298:
1294:
1290:
1285:
1281:
1273:
1268:
1261:
1257:
1253:
1248:
1243:
1236:
1232:
1225:
1218:
1213:
1206:
1200:
1197:
1195:
1190:
1185:
1177:
1171:
1163:
1162:
1136:
1135:
1123:
1119:
1114:
1109:
1104:
1099:
1094:
1089:
1083:
1079:
1075:
1070:
1065:
1060:
1055:
1050:
1045:
1028:
1021:
1014:
1007:
989:kinetic energy
984:
981:
980:
979:
968:
965:
962:
959:
956:
938:speed of light
908:
907:
896:
891:
886:
881:
878:
873:
868:
863:
858:
851:
847:
843:
838:
833:
828:
819:
815:
800:invariant mass
796:
795:
784:
777:
773:
767:
763:
757:
752:
748:
733:invariant mass
725:speed of light
698:
695:
687:minimal energy
679:
678:
667:
664:
659:
655:
649:
645:
639:
635:
631:
626:
622:
616:
612:
606:
602:
598:
590:
586:
580:
576:
568:
564:
558:
554:
548:
544:
535:
531:
525:
521:
517:
512:
508:
502:
498:
492:
488:
484:
481:
476:
472:
468:
463:
459:
455:
450:
446:
440:
436:
432:
428:
424:
420:
414:
410:
404:
400:
396:
392:
388:
384:
378:
374:
359:
358:
342:
338:
332:
328:
320:
316:
310:
306:
300:
296:
289:
284:
280:
265:
264:
253:
248:
244:
240:
237:
234:
230:
227:
195:
192:
190:
187:
175:center of mass
165:
152:inertial frame
130:
129:
112:September 2024
44:
42:
35:
28:
9:
6:
4:
3:
2:
3531:
3520:
3517:
3515:
3512:
3510:
3507:
3505:
3502:
3500:
3497:
3495:
3492:
3491:
3489:
3477:
3473:
3469:
3464:
3458:
3457:0-07-084018-0
3454:
3448:
3442:
3438:
3432:
3430:
3425:
3415:
3412:
3410:
3407:
3406:
3386:
3375:
3370:
3360:
3355:
3345:
3330:
3329:
3328:
3326:
3299:
3296:
3285:
3282:
3272:
3262:
3259:
3249:
3235:
3234:
3233:
3207:
3204:
3202:
3184:
3180:
3176:
3171:
3167:
3160:
3158:
3148:
3138:
3133:
3123:
3121:
3103:
3102:
3101:
3099:
3095:
3091:
3065:
3061:
3057:
3052:
3040:
3036:
3032:
3027:
3015:
3011:
3006:
2996:
2992:
2988:
2983:
2971:
2967:
2963:
2958:
2946:
2942:
2934:
2933:
2932:
2930:
2901:
2897:
2893:
2888:
2884:
2877:
2872:
2860:
2856:
2852:
2847:
2835:
2831:
2827:
2822:
2810:
2806:
2802:
2797:
2785:
2781:
2773:
2772:
2771:
2749:
2739:
2727:
2723:
2719:
2709:
2697:
2693:
2685:
2684:
2683:
2681:
2677:
2649:
2647:
2634:
2630:
2626:
2621:
2617:
2609:
2597:
2593:
2589:
2584:
2572:
2568:
2561:
2559:
2550:
2542:
2538:
2534:
2529:
2525:
2517:
2505:
2501:
2497:
2492:
2480:
2476:
2469:
2462:
2444:
2442:
2434:
2399:
2398:
2397:
2392:
2385:
2378:
2371:
2343:
2340:
2330:
2320:
2317:
2307:
2293:
2292:
2291:
2269:
2265:
2261:
2256:
2252:
2244:
2240:
2234:
2230:
2223:
2220:
2213:
2212:
2211:
2209:
2188:
2178:
2173:
2163:
2148:
2147:
2146:
2144:
2114:
2104:
2101:
2091:
2079:
2075:
2071:
2068:
2066:
2057:
2051:
2041:
2036:
2025:
2016:
2012:
2008:
2003:
1999:
1991:
1987:
1981:
1977:
1970:
1966:
1957:
1952:
1941:
1935:
1931:
1927:
1925:
1910:
1898:
1894:
1890:
1888:
1876:
1858:
1857:
1856:
1853:
1829:
1819:
1807:
1803:
1799:
1789:
1777:
1773:
1769:
1759:
1749:
1739:
1725:
1724:
1723:
1722:', vanishes:
1718:
1711:
1687:
1683:
1679:
1674:
1670:
1662:
1650:
1646:
1642:
1637:
1625:
1621:
1614:
1606:
1602:
1598:
1593:
1589:
1581:
1571:
1566:
1553:
1541:
1540:
1539:
1534:
1527:
1503:
1493:
1481:
1477:
1473:
1463:
1451:
1447:
1439:
1438:
1437:
1435:
1431:
1403:
1401:
1388:
1384:
1380:
1375:
1371:
1363:
1351:
1347:
1343:
1338:
1326:
1322:
1315:
1313:
1304:
1296:
1292:
1288:
1283:
1279:
1271:
1259:
1255:
1251:
1246:
1234:
1230:
1223:
1216:
1198:
1196:
1188:
1153:
1152:
1151:
1149:
1145:
1141:
1117:
1112:
1102:
1092:
1081:
1073:
1068:
1058:
1048:
1034:
1033:
1032:
1027:
1020:
1013:
1006:
1002:
998:
994:
990:
966:
963:
960:
957:
954:
947:
946:
945:
943:
939:
935:
932:
928:
924:
919:
917:
913:
894:
889:
884:
879:
876:
871:
866:
861:
856:
849:
845:
841:
836:
831:
826:
817:
813:
805:
804:
803:
801:
782:
775:
771:
765:
761:
755:
750:
746:
738:
737:
736:
734:
730:
726:
722:
718:
714:
713:
708:
704:
694:
692:
688:
684:
665:
662:
657:
653:
647:
643:
637:
633:
629:
624:
620:
614:
610:
604:
600:
596:
588:
584:
578:
574:
566:
562:
556:
552:
546:
542:
533:
529:
523:
519:
515:
510:
506:
500:
496:
490:
486:
482:
474:
470:
466:
461:
457:
448:
444:
438:
434:
430:
426:
422:
418:
412:
408:
402:
398:
394:
390:
386:
382:
376:
372:
364:
363:
362:
340:
336:
330:
326:
318:
314:
308:
304:
298:
294:
287:
282:
278:
270:
269:
268:
251:
246:
242:
238:
235:
232:
228:
225:
217:
216:
215:
213:
209:
205:
201:
186:
184:
180:
176:
172:
167:
163:
161:
157:
153:
149:
145:
141:
137:
126:
123:
115:
104:
101:
97:
94:
90:
87:
83:
80:
76:
73: –
72:
68:
67:Find sources:
61:
57:
51:
50:
45:This article
43:
39:
34:
33:
27:
19:
3467:
3463:
3447:
3322:
3231:
3097:
3093:
3089:
3087:
2928:
2926:
2769:
2679:
2675:
2673:
2390:
2383:
2376:
2369:
2366:
2289:
2208:reduced mass
2205:
2140:
1851:
1849:
1716:
1709:
1707:
1532:
1525:
1523:
1433:
1429:
1427:
1147:
1143:
1139:
1137:
1025:
1018:
1011:
1004:
988:
986:
920:
915:
911:
909:
797:
727:) gives the
720:
716:
710:
700:
686:
680:
360:
266:
211:
203:
199:
197:
170:
168:
159:
147:
143:
139:
133:
118:
109:
99:
92:
85:
78:
66:
54:Please help
49:verification
46:
26:
3414:Breit frame
2931:imply that
2141:notice the
712:rest energy
3514:Kinematics
3488:Categories
3420:References
703:relativity
189:Properties
82:newspapers
3379:Δ
3361:−
3338:Δ
3303:Δ
3300:μ
3289:Δ
3286:μ
3278:′
3263:−
3255:′
2745:′
2715:′
2347:Δ
2344:μ
2336:′
2321:−
2313:′
2221:μ
2179:−
2156:Δ
2120:′
2105:−
2097:′
2072:−
2042:−
1958:−
1916:′
1882:′
1825:′
1795:′
1765:′
1745:′
1499:′
1469:′
1118:−
1098:′
1074:−
1054:′
993:lab frame
925:(such as
923:rest mass
867:−
729:rest mass
634:∑
630:−
601:∑
575:∑
543:∑
520:∑
516:−
487:∑
467:−
435:∑
399:∑
373:∑
327:∑
295:∑
239:−
150:, is the
144:COM frame
3519:Momentum
3403:See also
942:massless
719:, where
427:′
391:′
229:′
156:momentum
927:photons
723:is the
267:where
194:General
136:physics
96:scholar
3474:
3455:
3439:
1715:' and
1138:where
683:energy
138:, the
98:
91:
84:
77:
69:
931:plane
214:′ is
164:frame
103:JSTOR
89:books
3472:ISBN
3453:ISBN
3437:ISBN
2389:and
2375:and
1531:and
1024:and
1010:and
999:and
940:, a
798:The
179:rest
75:news
2929:not
2210:is
701:In
166:".
134:In
58:by
3490::
3428:^
2678:=
1538::
1432:=
1430:R'
693:.
666:0.
3387:.
3383:u
3376:=
3371:2
3366:v
3356:1
3351:v
3346:=
3342:v
3307:u
3297:=
3293:v
3283:=
3273:2
3268:p
3260:=
3250:1
3245:p
3212:V
3208:M
3205:=
3194:V
3190:)
3185:2
3181:m
3177:+
3172:1
3168:m
3164:(
3161:=
3149:2
3144:p
3139:+
3134:1
3129:p
3124:=
3116:P
3098:P
3094:V
3090:M
3072:V
3066:2
3062:m
3058:=
3053:2
3048:v
3041:2
3037:m
3033:=
3028:2
3023:u
3016:2
3012:m
3007:,
3003:V
2997:1
2993:m
2989:=
2984:1
2979:v
2972:1
2968:m
2964:=
2959:1
2954:u
2947:1
2943:m
2911:V
2907:)
2902:2
2898:m
2894:+
2889:1
2885:m
2881:(
2878:=
2873:2
2868:v
2861:2
2857:m
2853:+
2848:1
2843:v
2836:1
2832:m
2828:=
2823:2
2818:u
2811:2
2807:m
2803:+
2798:1
2793:u
2786:1
2782:m
2754:0
2750:=
2740:2
2735:v
2728:2
2724:m
2720:+
2710:1
2705:v
2698:1
2694:m
2680:0
2676:R
2654:V
2650:=
2635:2
2631:m
2627:+
2622:1
2618:m
2610:2
2605:v
2598:2
2594:m
2590:+
2585:1
2580:v
2573:1
2569:m
2562:=
2551:)
2543:2
2539:m
2535:+
2530:1
2526:m
2518:2
2513:r
2506:2
2502:m
2498:+
2493:1
2488:r
2481:1
2477:m
2470:(
2463:t
2458:d
2451:d
2445:=
2435:t
2430:d
2422:R
2416:d
2394:2
2391:u
2387:1
2384:u
2380:2
2377:v
2373:1
2370:v
2351:u
2341:=
2331:2
2326:p
2318:=
2308:1
2303:p
2270:2
2266:m
2262:+
2257:1
2253:m
2245:2
2241:m
2235:1
2231:m
2224:=
2189:2
2184:u
2174:1
2169:u
2164:=
2160:u
2115:2
2110:p
2102:=
2092:2
2087:u
2080:2
2076:m
2069:=
2058:)
2052:2
2047:u
2037:1
2032:u
2026:(
2017:2
2013:m
2009:+
2004:1
2000:m
1992:2
1988:m
1982:1
1978:m
1971:=
1967:)
1962:V
1953:1
1948:u
1942:(
1936:1
1932:m
1928:=
1911:1
1906:u
1899:1
1895:m
1891:=
1877:1
1872:p
1852:V
1834:0
1830:=
1820:2
1815:u
1808:2
1804:m
1800:+
1790:1
1785:u
1778:1
1774:m
1770:=
1760:2
1755:p
1750:+
1740:1
1735:p
1720:2
1717:p
1713:1
1710:p
1688:2
1684:m
1680:+
1675:1
1671:m
1663:2
1658:u
1651:2
1647:m
1643:+
1638:1
1633:u
1626:1
1622:m
1615:=
1607:2
1603:m
1599:+
1594:1
1590:m
1582:2
1577:p
1572:+
1567:1
1562:p
1554:=
1550:V
1536:2
1533:p
1529:1
1526:p
1508:0
1504:=
1494:2
1489:u
1482:2
1478:m
1474:+
1464:1
1459:u
1452:1
1448:m
1434:0
1408:V
1404:=
1389:2
1385:m
1381:+
1376:1
1372:m
1364:2
1359:u
1352:2
1348:m
1344:+
1339:1
1334:u
1327:1
1323:m
1316:=
1305:)
1297:2
1293:m
1289:+
1284:1
1280:m
1272:2
1267:r
1260:2
1256:m
1252:+
1247:1
1242:r
1235:1
1231:m
1224:(
1217:t
1212:d
1205:d
1199:=
1189:t
1184:d
1176:R
1170:d
1148:R
1144:V
1140:V
1122:V
1113:2
1108:u
1103:=
1093:2
1088:u
1082:,
1078:V
1069:1
1064:u
1059:=
1049:1
1044:u
1029:2
1026:u
1022:1
1019:u
1015:2
1012:m
1008:1
1005:m
967:.
964:c
961:p
958:=
955:E
916:c
914:/
912:p
895:,
890:2
885:)
880:c
877:p
872:(
862:2
857:)
850:2
846:c
842:E
837:(
832:=
827:2
818:0
814:m
783:.
776:2
772:c
766:0
762:E
756:=
751:0
747:m
731:(
721:c
717:c
663:=
658:j
654:v
648:j
644:m
638:j
625:i
621:v
615:i
611:m
605:i
597:=
589:j
585:m
579:j
567:j
563:v
557:j
553:m
547:j
534:i
530:m
524:i
511:i
507:v
501:i
497:m
491:i
483:=
480:)
475:c
471:V
462:i
458:v
454:(
449:i
445:m
439:i
431:=
423:i
419:v
413:i
409:m
403:i
395:=
387:i
383:p
377:i
341:i
337:m
331:i
319:i
315:v
309:i
305:m
299:i
288:=
283:c
279:V
252:,
247:c
243:V
236:v
233:=
226:v
212:S
204:S
200:S
142:(
125:)
119:(
114:)
110:(
100:·
93:·
86:·
79:·
52:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
