4988:
4481:
4983:{\displaystyle \mathbf {\hat {t}} =\left\{{\begin{matrix}{\frac {\mathbf {v} _{r}-(\mathbf {v} _{r}\cdot \mathbf {\hat {n}} )\mathbf {\hat {n}} }{|\mathbf {v} _{r}-(\mathbf {v} _{r}\cdot \mathbf {\hat {n}} )\mathbf {\hat {n}} |}}&\mathbf {v} _{r}\cdot \mathbf {\hat {n}} \neq 0&\\{\frac {\mathbf {f} _{e}-(\mathbf {f} _{e}\cdot \mathbf {\hat {n}} )\mathbf {\hat {n}} }{|\mathbf {f} _{e}-(\mathbf {f} _{e}\cdot \mathbf {\hat {n}} )\mathbf {\hat {n}} |}}&\mathbf {v} _{r}\cdot \mathbf {\hat {n}} =0&\mathbf {f} _{e}\cdot \mathbf {\hat {n}} \neq 0\\\mathbf {0} &\mathbf {v} _{r}\cdot \mathbf {\hat {n}} =0&\mathbf {f} _{e}\cdot \mathbf {\hat {n}} =0\\\end{matrix}}\right.}
941:
3881:
265:
3123:
6140:
5505:
22:
2827:
174:, may be traced to the behaviour of real bodies that, unlike their perfectly rigid idealised counterparts, do undergo minor compression on collision, followed by expansion, prior to separation. The compression phase converts the kinetic energy of the bodies into potential energy and to an extent, heat. The expansion phase converts the potential energy back to kinetic energy.
5931:
5302:
146:
746:, that is, a smaller time interval must be compensated by a stronger reaction force to achieve the same impulse. When modelling a collision between idealized rigid bodies, it is impractical to simulate the compression and expansion phases of the body geometry over the collision time interval. However, by assuming that a force
3118:{\displaystyle j_{r}={\frac {-(1+e)\mathbf {v} _{r}\cdot \mathbf {\hat {n}} }{{m_{1}}^{-1}+{m_{2}}^{-1}+({\mathbf {I} _{1}}^{-1}(\mathbf {r} _{1}\times \mathbf {\hat {n}} )\times \mathbf {r} _{1}+{\mathbf {I} _{2}}^{-1}(\mathbf {r} _{2}\times \mathbf {\hat {n}} )\times \mathbf {r} _{2})\cdot \mathbf {\hat {n}} }}}
276:
To overcome the friction between two bodies in static contact, the surfaces must somehow lift away from each other. Once in motion, the degree of surface affinity is reduced and hence bodies in sliding motion tend to offer lesser resistance to motion. These two categories of friction are respectively
226:
of the relative post-collision speed of a point of contact along the contact normal, with respect to the relative pre-collision speed of the same point along the same normal. These coefficients are typically determined empirically for different material pairs, such as wood against concrete or rubber
177:
During the compression and expansion phases of two colliding bodies, each body generates reactive forces on the other at the points of contact, such that the sum reaction forces of one body are equal in magnitude but opposite in direction to the forces of the other, as per the
Newtonian principle of
161:
techniques. On collision, the kinetic properties of two such bodies seem to undergo an instantaneous change, typically resulting in the bodies rebounding away from each other, sliding, or settling into relative static contact, depending on the elasticity of the materials and the configuration of the
6135:{\displaystyle \mathbf {j} _{f}=\left\{{\begin{matrix}-(m\mathbf {v} _{r}\cdot \mathbf {\hat {t}} )\mathbf {\hat {t}} &\mathbf {v} _{r}\cdot \mathbf {\hat {t}} =0&m\mathbf {v} _{r}\cdot \mathbf {\hat {t}} \leq j_{s}\\-j_{d}\mathbf {\hat {t}} &{\text{(otherwise)}}\\\end{matrix}}\right.}
251:
such as a piece of soft clay hitting the floor, whereas values close to one represent highly elastic collisions, such as a rubber ball bouncing off a wall. The kinetic energy loss is relative to one body with respect to the other. Thus the total momentum of both bodies with respect to some common
4270:
defines a maximum magnitude for the friction force required to counter the tangential component of any external sum force applied on a relatively static body, such that it remains static. Thus, if the external force is large enough, static friction is unable to fully counter this force, at which
5500:{\displaystyle \mathbf {f} _{f}=\left\{{\begin{matrix}-(\mathbf {f} _{e}\cdot \mathbf {\hat {t}} )\mathbf {\hat {t}} &\mathbf {v} _{r}\cdot \mathbf {\hat {t}} =0&\mathbf {f} _{e}\cdot \mathbf {\hat {t}} \leq f_{s}\\-f_{d}\mathbf {\hat {t}} &{\text{(otherwise)}}\\\end{matrix}}\right.}
313:
It is the force exerted by a surface as an object moves across it or makes an effort to move across it. The friction force opposes the motion of the object. Friction results when two surfaces are pressed together closely, causing attractive intermolecular forces between the molecules of the two
272:
Another important contact phenomenon is surface-to-surface friction, a force that impedes the relative motion of two surfaces in contact, or that of a body in a fluid. In this section we discuss surface-to-surface friction of two bodies in relative static contact or sliding contact. In the real
4305:
within which the tangential component of a force exerted by one body on the surface of another in static contact, is countered by an equal and opposite force such that the static configuration is maintained. Conversely, if the force falls outside the cone, static friction gives way to dynamic
314:
different surface. As such, friction depends upon the nature of the two surfaces and upon the degree to which they are pressed together. Friction always acts parallel to the surface in contact and opposite the direction of motion. The friction force can be calculated using the equation.
2065:
1942:
2780:
5516:
Equations (6a), (6b), (7) and (8) describe the
Coulomb friction model in terms of forces. By adapting the argument for instantaneous impulses, an impulse-based version of the Coulomb friction model may be derived, relating a frictional impulse
1696:
1600:
2454:
744:
2659:
178:
action and reaction. If the effects of friction are ignored, a collision is seen as affecting only the component of the velocities that are directed along the contact normal and as leaving the tangential components unaffected
902:
1179:
6253:
C. Vella, "Gravitas: An extensible physics engine framework using object-oriented and design pattern-driven software architecture principles," Master in
Information Technology Thesis, University of Malta, Msida,
2579:
1346:
658:
2236:
1955:
1832:
1817:
3625:
4439:
4351:
2186:
1098:
999:
5632:
5559:
5289:
5216:
5114:
5041:
4395:
2284:
1768:
1225:
366:
305:
is sometimes referred to as the pressing force since its action presses the surface together. Normal force is always directed towards the object and acts perpendicularly with the applied force.
2370:
2137:
591:
519:
3784:
3204:
4062:
468:
4016:. These coefficients describe the two types of friction forces in terms of the reaction forces acting on the bodies. More specifically, the static and dynamic friction force magnitudes
3970:
3930:
5116:
for both the general and particular states of contact. Informally, the first case computes the tangent vector along the relative velocity component perpendicular to the contact normal
5860:
4114:
4014:
3813:
3662:
3170:
5800:
5740:
4233:
4173:
1381:
1442:
5918:
5889:
5588:
5245:
5172:
5143:
5070:
4468:
3571:
3463:
2688:
1471:
1410:
1286:
422:
394:
3842:
3747:
3691:
3515:
3434:
3405:
3376:
3347:
3264:
1444:
allows for the derivation of a formula to compute the change in linear and angular velocities of the bodies resulting from the collision impulses. In the subsequent formulas,
1254:
550:
2700:
2306:
926:
766:
682:
6207:
6178:
273:
world, friction is due to the imperfect microstructure of surfaces whose protrusions interlock into each other, generating reactive forces tangential to the surfaces.
2382:
6151:
Equations (5) and (10) define an impulse-based contact model that is ideal for impulse-based simulations. When using this model, care must be taken in the choice of
4296:
4268:
3869:
3718:
3542:
3318:
3291:
3235:
2815:
2494:
1501:
813:
687:
224:
1613:
1517:
3483:
2514:
2093:
1724:
786:
245:
5681:
1045:
820:
293:
It is a force which is applied to an object by another object or by a person. The direction of the applied force depends on how the force is applied.
1183:
By deduction from the principle of action and reaction, if the collision impulse applied by the first body on the second body at a contact point
6252:
1105:
137:
deals with models and algorithms for simulating the changes in the motion of two solid bodies following collision and other forms of contact.
2584:
2519:
39:
1291:
598:
2060:{\displaystyle \mathbf {\omega '} _{2}=\mathbf {\omega } _{2}+j_{r}\mathbf {I} _{2}^{-1}(\mathbf {r} _{2}\times \mathbf {\hat {n}} )}
1937:{\displaystyle \mathbf {\omega '} _{1}=\mathbf {\omega } _{1}-j_{r}\mathbf {I} _{1}^{-1}(\mathbf {r} _{1}\times \mathbf {\hat {n}} )}
105:
2191:
86:
3576:
58:
4400:
4312:
1050:
951:
43:
157:
in unconstrained motion, potentially under the action of forces, may be modelled by solving their equations of motion using
1773:
5593:
5520:
5250:
5177:
5075:
5002:
4356:
2245:
1729:
1186:
65:
325:
2142:
2790:
Substituting equations (1a), (1b), (2a), (2b) and (3) into equation (4) and solving for the reaction impulse magnitude
2314:
2098:
2372:
of the bodies at the point of contact may be computed in terms of the respective linear and angular velocities, using
559:
473:
72:
1504:
191:
32:
4019:
427:
3938:
3898:
940:
54:
5813:
4067:
3975:
3755:
3175:
3789:
5753:
5693:
4186:
4126:
1351:
1415:
6238:
An
Introduction to Physically Based Modeling (David Baraff Robotics Institute Carnegie Mellon University)
5894:
5865:
5564:
5221:
5148:
5119:
5046:
4444:
3547:
3439:
2664:
1447:
1386:
158:
3633:
3141:
1259:
399:
371:
3818:
3723:
3667:
3491:
3410:
3381:
3352:
3323:
3240:
2775:{\displaystyle \mathbf {v'} _{r}\cdot \mathbf {\hat {n}} =-e\mathbf {v} _{r}\cdot \mathbf {\hat {n}} }
1230:
524:
3880:
6218:
4302:
264:
248:
122:
2289:
909:
749:
665:
6268:
6183:
6154:
2449:{\displaystyle \mathbf {v} _{pi}=\mathbf {v} _{i}+\mathbf {\omega } _{i}\times \mathbf {r} _{i}}
79:
6237:
301:
It is the support force exerted upon an object which is in contact with another stable object.
739:{\displaystyle t_{1}\rightarrow t_{0}\Rightarrow \left|\mathbf {f} \right|\rightarrow \infty }
3933:
3889:
1691:{\displaystyle \mathbf {v'} _{2}=\mathbf {v} _{2}+{\frac {j_{r}}{m_{2}}}\mathbf {\hat {n}} }
1595:{\displaystyle \mathbf {v'} _{1}=\mathbf {v} _{1}-{\frac {j_{r}}{m_{1}}}\mathbf {\hat {n}} }
553:
4274:
4246:
3847:
3696:
3520:
3296:
3269:
3213:
2793:
2467:
1479:
791:
197:
5218:. If there is no tangential velocity or external forces, then no friction is assumed, and
3893:
126:
3468:
2499:
2239:
2078:
1709:
771:
230:
5637:
1004:
145:
6262:
1473:
is always assumed to point away from body 1 and towards body 2 at the contact point.
932:
may be introduced to simulate an instantaneous change in velocity after a collision.
4271:
point the body gains velocity and becomes subject to dynamic friction of magnitude
302:
154:
1507:
the relation between the bodies' pre- and post- linear velocities are as follows
897:{\displaystyle \lim _{t_{1}\rightarrow t_{0}}\int _{t_{0}}^{t_{1}}\mathbf {f} dt}
130:
21:
186:
The degree of relative kinetic energy retained after a collision, termed the
5043:
is the sum of all external forces on the body. The multi-case definition of
149:
The compression and expansion phases of a collision between two solid bodies
6209:
as higher values may introduce additional kinetic energy into the system.
252:
reference is unchanged after the collision, in line with the principle of
1174:{\displaystyle \mathbf {j} _{r}=\int _{t_{0}}^{t_{1}}\mathbf {f} _{r}dt}
3138:
Thus, the procedure for computing the post-collision linear velocities
2654:{\displaystyle \mathbf {v'} _{r}=\mathbf {v'} _{p2}-\mathbf {v'} _{p1}}
552:
is the resulting change in velocity. The change in momentum, termed an
5920:
constant throughout the time interval, the integration of (8) yields
5862:
is the magnitude of the reaction impulse acting along contact normal
2574:{\displaystyle \mathbf {v} _{r}=\mathbf {v} _{p2}-\mathbf {v} _{p1}}
3879:
939:
263:
144:
1341:{\displaystyle \pm \mathbf {j} _{r}=\pm j_{r}\mathbf {\hat {n}} }
1256:, the counter impulse applied by the second body on the first is
653:{\displaystyle \mathbf {j} =\int _{t_{0}}^{t_{1}}\mathbf {f} dt}
190:, is dependent on the elasticity of the bodies‟ materials. The
3888:
One of the most popular models for describing friction is the
1047:
may hence be represented by an instantaneous reaction impulse
15:
5634:. Integrating (6a) and (6b) over the collision time interval
5072:
is required for robustly computing the actual friction force
2581:
of the contact point to the post-collision relative velocity
2231:{\displaystyle \mathbf {I} _{i}\in \mathbb {R} ^{3\times 3}}
6129:
5494:
4977:
118:
Models and algorithms for simulating collision and reaction
3620:{\displaystyle \mathbf {j} _{r}=j_{r}\mathbf {\hat {n}} }
5174:
in terms of the tangent component of the external force
4434:{\displaystyle \mathbf {\hat {t}} \in \mathbb {R} ^{3}}
4346:{\displaystyle \mathbf {\hat {n}} \in \mathbb {R} ^{3}}
1093:{\displaystyle \mathbf {j} _{r}(t)\in \mathbb {R} ^{3}}
994:{\displaystyle \mathbf {f} _{r}(t)\in \mathbb {R} ^{3}}
5955:
5326:
4505:
4064:
are computed in terms of the reaction force magnitude
6186:
6157:
5934:
5897:
5868:
5816:
5756:
5696:
5640:
5596:
5567:
5523:
5305:
5253:
5224:
5180:
5151:
5145:. If this component is zero, the second case derives
5122:
5078:
5049:
5005:
4484:
4447:
4403:
4359:
4315:
4277:
4249:
4189:
4129:
4070:
4022:
3978:
3941:
3901:
3850:
3821:
3792:
3758:
3726:
3699:
3670:
3636:
3579:
3550:
3523:
3494:
3471:
3442:
3413:
3384:
3355:
3326:
3299:
3272:
3243:
3216:
3178:
3144:
2830:
2796:
2703:
2667:
2587:
2522:
2502:
2470:
2385:
2317:
2292:
2248:
2194:
2145:
2101:
2081:
1958:
1835:
1812:{\displaystyle \mathbf {v'} _{i}\in \mathbb {R} ^{3}}
1776:
1732:
1712:
1616:
1520:
1482:
1450:
1418:
1389:
1354:
1294:
1262:
1233:
1189:
1108:
1053:
1007:
954:
944:
The application of impulses at the point of collision
912:
823:
794:
774:
752:
690:
668:
601:
562:
527:
476:
430:
402:
374:
328:
233:
200:
5627:{\displaystyle \mathbf {j} _{r}\in \mathbb {R} ^{3}}
5554:{\displaystyle \mathbf {j} _{f}\in \mathbb {R} ^{3}}
5284:{\displaystyle \mathbf {f} _{f}\in \mathbb {R} ^{3}}
5211:{\displaystyle \mathbf {f} _{e}\in \mathbb {R} ^{3}}
5109:{\displaystyle \mathbf {f} _{f}\in \mathbb {R} ^{3}}
5036:{\displaystyle \mathbf {f} _{e}\in \mathbb {R} ^{3}}
4390:{\displaystyle \mathbf {v} _{r}\in \mathbb {R} ^{3}}
2279:{\displaystyle \mathbf {r} _{i}\in \mathbb {R} ^{3}}
1763:{\displaystyle \mathbf {v} _{i}\in \mathbb {R} ^{3}}
1220:{\displaystyle \mathbf {p} _{r}\in \mathbb {R} ^{3}}
268:
Friction due to surface microstructure imperfections
194:
between two given materials is modeled as the ratio
361:{\displaystyle \mathbf {f} (t)\in \mathbb {R} ^{3}}
46:. Unsourced material may be challenged and removed.
6201:
6172:
6134:
5912:
5883:
5854:
5794:
5734:
5675:
5626:
5582:
5553:
5499:
5283:
5239:
5210:
5166:
5137:
5108:
5064:
5035:
4982:
4462:
4433:
4389:
4345:
4290:
4262:
4227:
4167:
4108:
4056:
4008:
3964:
3924:
3863:
3836:
3807:
3778:
3741:
3712:
3685:
3656:
3619:
3565:
3536:
3509:
3477:
3457:
3428:
3399:
3370:
3341:
3312:
3285:
3258:
3229:
3198:
3164:
3117:
2809:
2774:
2682:
2653:
2573:
2508:
2488:
2448:
2364:
2300:
2278:
2230:
2181:{\displaystyle {\omega '}_{i}\in \mathbb {R} ^{3}}
2180:
2131:
2087:
2059:
1936:
1811:
1762:
1718:
1690:
1594:
1495:
1465:
1436:
1404:
1375:
1340:
1280:
1248:
1219:
1173:
1092:
1039:
993:
920:
896:
807:
780:
760:
738:
676:
652:
585:
544:
513:
462:
416:
388:
360:
239:
218:
6112:
6070:
6032:
6003:
5988:
5904:
5875:
5574:
5477:
5435:
5400:
5371:
5356:
5231:
5158:
5129:
5056:
4961:
4926:
4882:
4847:
4810:
4795:
4743:
4728:
4669:
4632:
4617:
4565:
4550:
4491:
4454:
4410:
4322:
4301:The Coulomb friction model effectively defines a
3611:
3557:
3449:
3106:
3070:
2995:
2886:
2766:
2730:
2674:
2365:{\displaystyle v_{p1},v_{p2}\in \mathbb {R} ^{3}}
2132:{\displaystyle {\omega }_{i}\in \mathbb {R} ^{3}}
2048:
1925:
1682:
1586:
1457:
1428:
1396:
1332:
825:
586:{\displaystyle \mathbf {j} \in \mathbb {R} ^{3}}
514:{\displaystyle \mathbf {p} (t)=m\mathbf {v} (t)}
8:
2516:relates the pre-collision relative velocity
457:
431:
396:, acting on a body of assumed constant mass
4057:{\displaystyle f_{s},f_{d}\in \mathbb {R} }
463:{\displaystyle \lbrack t_{0},t_{1}\rbrack }
3965:{\displaystyle {\mu }_{d}\in \mathbb {R} }
3925:{\displaystyle {\mu }_{s}\in \mathbb {R} }
3892:model. This model defines coefficients of
470:generates a change in the body’s momentum
6193:
6188:
6185:
6164:
6159:
6156:
6120:
6107:
6106:
6100:
6083:
6065:
6064:
6055:
6050:
6027:
6026:
6017:
6012:
5998:
5997:
5983:
5982:
5973:
5968:
5954:
5941:
5936:
5933:
5899:
5898:
5896:
5870:
5869:
5867:
5847:
5841:
5836:
5830:
5821:
5815:
5786:
5776:
5771:
5761:
5755:
5726:
5716:
5711:
5701:
5695:
5664:
5648:
5639:
5618:
5614:
5613:
5603:
5598:
5595:
5569:
5568:
5566:
5545:
5541:
5540:
5530:
5525:
5522:
5485:
5472:
5471:
5465:
5448:
5430:
5429:
5420:
5415:
5395:
5394:
5385:
5380:
5366:
5365:
5351:
5350:
5341:
5336:
5325:
5312:
5307:
5304:
5275:
5271:
5270:
5260:
5255:
5252:
5226:
5225:
5223:
5202:
5198:
5197:
5187:
5182:
5179:
5153:
5152:
5150:
5124:
5123:
5121:
5100:
5096:
5095:
5085:
5080:
5077:
5051:
5050:
5048:
5027:
5023:
5022:
5012:
5007:
5004:
4956:
4955:
4946:
4941:
4921:
4920:
4911:
4906:
4898:
4877:
4876:
4867:
4862:
4842:
4841:
4832:
4827:
4816:
4805:
4804:
4790:
4789:
4780:
4775:
4762:
4757:
4751:
4738:
4737:
4723:
4722:
4713:
4708:
4695:
4690:
4686:
4664:
4663:
4654:
4649:
4638:
4627:
4626:
4612:
4611:
4602:
4597:
4584:
4579:
4573:
4560:
4559:
4545:
4544:
4535:
4530:
4517:
4512:
4508:
4504:
4486:
4485:
4483:
4449:
4448:
4446:
4425:
4421:
4420:
4405:
4404:
4402:
4381:
4377:
4376:
4366:
4361:
4358:
4337:
4333:
4332:
4317:
4316:
4314:
4282:
4276:
4254:
4248:
4219:
4209:
4204:
4194:
4188:
4159:
4149:
4144:
4134:
4128:
4101:
4095:
4090:
4084:
4075:
4069:
4050:
4049:
4040:
4027:
4021:
4000:
3995:
3985:
3980:
3977:
3958:
3957:
3948:
3943:
3940:
3918:
3917:
3908:
3903:
3900:
3855:
3849:
3828:
3823:
3820:
3799:
3794:
3791:
3770:
3760:
3757:
3733:
3728:
3725:
3704:
3698:
3677:
3672:
3669:
3648:
3638:
3635:
3606:
3605:
3599:
3586:
3581:
3578:
3552:
3551:
3549:
3528:
3522:
3501:
3496:
3493:
3470:
3444:
3443:
3441:
3420:
3415:
3412:
3391:
3386:
3383:
3362:
3357:
3354:
3333:
3328:
3325:
3304:
3298:
3277:
3271:
3250:
3245:
3242:
3221:
3215:
3190:
3180:
3177:
3156:
3146:
3143:
3101:
3100:
3088:
3083:
3065:
3064:
3055:
3050:
3037:
3030:
3025:
3023:
3013:
3008:
2990:
2989:
2980:
2975:
2962:
2955:
2950:
2948:
2932:
2925:
2920:
2907:
2900:
2895:
2881:
2880:
2871:
2866:
2844:
2835:
2829:
2801:
2795:
2761:
2760:
2751:
2746:
2725:
2724:
2715:
2705:
2702:
2669:
2668:
2666:
2642:
2632:
2619:
2609:
2599:
2589:
2586:
2562:
2557:
2544:
2539:
2529:
2524:
2521:
2501:
2469:
2440:
2435:
2425:
2420:
2410:
2405:
2392:
2387:
2384:
2356:
2352:
2351:
2338:
2322:
2316:
2293:
2291:
2270:
2266:
2265:
2255:
2250:
2247:
2216:
2212:
2211:
2201:
2196:
2193:
2172:
2168:
2167:
2157:
2147:
2144:
2123:
2119:
2118:
2108:
2103:
2100:
2080:
2043:
2042:
2033:
2028:
2015:
2010:
2005:
1998:
1985:
1980:
1970:
1960:
1957:
1920:
1919:
1910:
1905:
1892:
1887:
1882:
1875:
1862:
1857:
1847:
1837:
1834:
1803:
1799:
1798:
1788:
1778:
1775:
1754:
1750:
1749:
1739:
1734:
1731:
1711:
1677:
1676:
1668:
1658:
1652:
1643:
1638:
1628:
1618:
1615:
1581:
1580:
1572:
1562:
1556:
1547:
1542:
1532:
1522:
1519:
1487:
1481:
1476:Assuming the collision impulse magnitude
1452:
1451:
1449:
1423:
1422:
1417:
1391:
1390:
1388:
1369:
1368:
1359:
1353:
1327:
1326:
1320:
1304:
1299:
1293:
1272:
1267:
1261:
1240:
1235:
1232:
1211:
1207:
1206:
1196:
1191:
1188:
1159:
1154:
1145:
1140:
1133:
1128:
1115:
1110:
1107:
1084:
1080:
1079:
1060:
1055:
1052:
1028:
1015:
1006:
985:
981:
980:
961:
956:
953:
913:
911:
883:
875:
870:
863:
858:
846:
833:
828:
822:
799:
793:
773:
753:
751:
721:
708:
695:
689:
669:
667:
639:
631:
626:
619:
614:
602:
600:
577:
573:
572:
563:
561:
528:
526:
497:
477:
475:
451:
438:
429:
410:
409:
401:
382:
381:
373:
352:
348:
347:
329:
327:
232:
199:
170:The origin of the rebound phenomenon, or
106:Learn how and when to remove this message
5927:
5855:{\displaystyle j_{r}=|\mathbf {j} _{r}|}
5750:
5690:
5298:
4477:
4183:
4123:
4109:{\displaystyle f_{r}=|\mathbf {f} _{r}|}
4009:{\displaystyle {\mu }_{s}>{\mu }_{d}}
3126:
2824:
2819:
2783:
2697:
2692:
2457:
2379:
2374:
2188:is the angular post-collision velocity,
2068:
1952:
1945:
1829:
1824:
1699:
1610:
1603:
1514:
1509:
6230:
6144:
5922:
5803:
5743:
5685:
5509:
5293:
4992:
4472:
4397:of the contact point, a tangent vector
4236:
4176:
4118:
3779:{\displaystyle \mathbf {\omega '} _{i}}
3210:Compute the reaction impulse magnitude
3199:{\displaystyle \mathbf {\omega '} _{i}}
2139:is the angular pre-collision velocity,
1819:is the post-collision linear velocity.
1383:and direction along the contact normal
5247:may be set to the zero vector. Thus,
3884:Coulomb friction model - friction cone
3808:{\displaystyle \mathbf {\omega } _{i}}
2286:is offset of the shared contact point
1770:is the pre-collision linear velocity,
5795:{\displaystyle j_{d}={\mu }_{d}j_{r}}
5735:{\displaystyle j_{s}={\mu }_{s}j_{r}}
4298:acting against the sliding velocity.
4228:{\displaystyle f_{d}={\mu }_{d}f_{r}}
4168:{\displaystyle f_{s}={\mu }_{s}f_{r}}
2242:in the world frame of reference, and
1822:Similarly for the angular velocities
1376:{\displaystyle j_{r}\in \mathbb {R} }
7:
3488:Compute the reaction impulse vector
1437:{\displaystyle -\mathbf {\hat {n}} }
44:adding citations to reliable sources
5913:{\displaystyle \mathbf {\hat {t}} }
5884:{\displaystyle \mathbf {\hat {n}} }
5583:{\displaystyle \mathbf {\hat {t}} }
5240:{\displaystyle \mathbf {\hat {t}} }
5167:{\displaystyle \mathbf {\hat {t}} }
5138:{\displaystyle \mathbf {\hat {n}} }
5065:{\displaystyle \mathbf {\hat {t}} }
4463:{\displaystyle \mathbf {\hat {n}} }
3786:in terms of old angular velocities
3566:{\displaystyle \mathbf {\hat {n}} }
3458:{\displaystyle \mathbf {\hat {n}} }
2683:{\displaystyle \mathbf {\hat {n}} }
1466:{\displaystyle \mathbf {\hat {n}} }
1405:{\displaystyle \mathbf {\hat {n}} }
948:The effect of the reaction force
733:
14:
3657:{\displaystyle \mathbf {v'} _{i}}
3165:{\displaystyle \mathbf {v'} _{i}}
2496:. The coefficient of restitution
1281:{\displaystyle -\mathbf {j} _{r}}
417:{\displaystyle m\in \mathbb {R} }
389:{\displaystyle t\in \mathbb {R} }
6109:
6067:
6051:
6029:
6013:
6000:
5985:
5969:
5937:
5901:
5872:
5837:
5599:
5571:
5526:
5474:
5432:
5416:
5397:
5381:
5368:
5353:
5337:
5308:
5256:
5228:
5183:
5155:
5126:
5081:
5053:
5008:
4958:
4942:
4923:
4907:
4899:
4879:
4863:
4844:
4828:
4807:
4792:
4776:
4758:
4740:
4725:
4709:
4691:
4666:
4650:
4629:
4614:
4598:
4580:
4562:
4547:
4531:
4513:
4488:
4451:
4407:
4362:
4319:
4091:
3837:{\displaystyle \mathbf {I} _{i}}
3824:
3742:{\displaystyle \mathbf {j} _{r}}
3729:
3686:{\displaystyle \mathbf {v} _{i}}
3673:
3640:
3608:
3582:
3554:
3510:{\displaystyle \mathbf {j} _{r}}
3497:
3446:
3429:{\displaystyle \mathbf {r} _{2}}
3416:
3400:{\displaystyle \mathbf {r} _{1}}
3387:
3371:{\displaystyle \mathbf {I} _{2}}
3358:
3342:{\displaystyle \mathbf {I} _{1}}
3329:
3259:{\displaystyle \mathbf {v} _{r}}
3246:
3148:
3134:Computing impulse-based reaction
3103:
3084:
3067:
3051:
3026:
3009:
2992:
2976:
2951:
2883:
2867:
2763:
2747:
2727:
2707:
2671:
2634:
2611:
2591:
2558:
2540:
2525:
2436:
2406:
2388:
2294:
2251:
2197:
2045:
2029:
2006:
1922:
1906:
1883:
1780:
1735:
1679:
1639:
1620:
1583:
1543:
1524:
1454:
1425:
1393:
1329:
1300:
1268:
1249:{\displaystyle \mathbf {j} _{r}}
1236:
1192:
1155:
1111:
1056:
957:
914:
884:
754:
722:
670:
640:
603:
564:
529:
498:
478:
330:
20:
3752:Compute new angular velocities
1001:over the interval of collision
768:can be found which is equal to
545:{\displaystyle \mathbf {v} (t)}
31:needs additional citations for
5994:
5961:
5848:
5831:
5670:
5641:
5362:
5332:
4817:
4801:
4771:
4752:
4734:
4704:
4639:
4623:
4593:
4574:
4556:
4526:
4102:
4085:
3630:Compute new linear velocities
3094:
3076:
3046:
3001:
2971:
2944:
2862:
2850:
2054:
2024:
1931:
1901:
1072:
1066:
1034:
1008:
973:
967:
839:
730:
714:
701:
539:
533:
508:
502:
488:
482:
340:
334:
213:
207:
1:
3871:using equations (2a) and (2b)
3749:using equations (1a) and (1b)
684:, the equation suggests that
3876:Impulse-based friction model
3720:and reaction impulse vector
2301:{\displaystyle \mathbf {p} }
936:Impulse-based reaction model
921:{\displaystyle \mathbf {j} }
761:{\displaystyle \mathbf {f} }
677:{\displaystyle \mathbf {j} }
5561:, acting along the tangent
4470:, may be defined such that
3664:in terms of old velocities
1348:into the impulse magnitude
318:Impulse-based contact model
6285:
6202:{\displaystyle {\mu }_{d}}
6173:{\displaystyle {\mu }_{s}}
5590:, to the reaction impulse
3517:in terms of its magnitude
815:, and such that the limit
192:coefficient of restitution
5925:
5891:. Similarly, by assuming
5748:
5688:
5296:
4475:
4309:Given the contact normal
4181:
4121:
2822:
2695:
2661:along the contact normal
2377:
2308:from the centre of mass.
1950:
1827:
1608:
1512:
227:against wood. Values for
254:conservation of momentum
3172:and angular velocities
1505:Newton's laws of motion
906:exists and is equal to
247:close to zero indicate
6203:
6174:
6136:
5914:
5885:
5856:
5796:
5736:
5677:
5628:
5584:
5555:
5501:
5285:
5241:
5212:
5168:
5139:
5110:
5066:
5037:
4984:
4464:
4435:
4391:
4353:and relative velocity
4347:
4292:
4264:
4229:
4169:
4110:
4058:
4010:
3966:
3926:
3885:
3865:
3838:
3809:
3780:
3743:
3714:
3687:
3658:
3621:
3567:
3538:
3511:
3479:
3459:
3430:
3401:
3372:
3343:
3314:
3287:
3260:
3231:
3200:
3166:
3119:
2811:
2776:
2684:
2655:
2575:
2510:
2490:
2450:
2366:
2302:
2280:
2232:
2182:
2133:
2089:
2061:
1938:
1813:
1764:
1720:
1692:
1596:
1497:
1467:
1438:
1406:
1377:
1342:
1282:
1250:
1221:
1175:
1094:
1041:
995:
945:
930:instantaneous impulses
922:
898:
809:
782:
762:
740:
678:
654:
587:
546:
515:
464:
418:
390:
362:
269:
241:
220:
150:
6204:
6175:
6137:
5915:
5886:
5857:
5797:
5737:
5678:
5629:
5585:
5556:
5502:
5286:
5242:
5213:
5169:
5140:
5111:
5067:
5038:
4985:
4465:
4436:
4392:
4348:
4293:
4291:{\displaystyle f_{d}}
4265:
4263:{\displaystyle f_{s}}
4230:
4170:
4111:
4059:
4011:
3967:
3927:
3883:
3866:
3864:{\displaystyle j_{r}}
3844:and reaction impulse
3839:
3810:
3781:
3744:
3715:
3713:{\displaystyle m_{i}}
3688:
3659:
3622:
3568:
3539:
3537:{\displaystyle j_{r}}
3512:
3480:
3460:
3431:
3402:
3373:
3344:
3315:
3313:{\displaystyle m_{2}}
3288:
3286:{\displaystyle m_{1}}
3261:
3232:
3230:{\displaystyle j_{r}}
3201:
3167:
3120:
2812:
2810:{\displaystyle j_{r}}
2777:
2685:
2656:
2576:
2511:
2491:
2489:{\displaystyle i=1,2}
2451:
2367:
2303:
2281:
2233:
2183:
2134:
2090:
2062:
1939:
1814:
1765:
1721:
1693:
1597:
1498:
1496:{\displaystyle j_{r}}
1468:
1439:
1407:
1378:
1343:
1283:
1251:
1222:
1176:
1095:
1042:
996:
943:
923:
899:
810:
808:{\displaystyle t_{0}}
788:everywhere except at
783:
763:
741:
679:
655:
588:
547:
516:
465:
419:
391:
363:
267:
242:
221:
219:{\displaystyle e\in }
159:numerical integration
148:
6184:
6155:
5932:
5895:
5866:
5814:
5754:
5694:
5638:
5594:
5565:
5521:
5303:
5251:
5222:
5178:
5149:
5120:
5076:
5047:
5003:
4482:
4445:
4401:
4357:
4313:
4275:
4247:
4187:
4127:
4068:
4020:
3976:
3939:
3899:
3848:
3819:
3790:
3756:
3724:
3697:
3668:
3634:
3577:
3548:
3521:
3492:
3469:
3440:
3411:
3382:
3353:
3324:
3297:
3270:
3241:
3214:
3176:
3142:
2828:
2794:
2701:
2665:
2585:
2520:
2500:
2468:
2383:
2315:
2290:
2246:
2192:
2143:
2099:
2079:
1956:
1833:
1774:
1730:
1710:
1614:
1518:
1480:
1448:
1416:
1387:
1352:
1292:
1288:. The decomposition
1260:
1231:
1187:
1106:
1051:
1005:
952:
910:
821:
792:
772:
750:
688:
666:
599:
593:is thus computed as
560:
525:
474:
428:
424:for a time interval
400:
372:
368:, dependent on time
326:
249:inelastic collisions
231:
198:
55:"Collision response"
40:improve this article
6219:Collision detection
3544:and contact normal
2023:
1900:
1503:is given and using
1152:
882:
638:
123:classical mechanics
6199:
6170:
6132:
6127:
5910:
5881:
5852:
5792:
5732:
5673:
5624:
5580:
5551:
5497:
5492:
5281:
5237:
5208:
5164:
5135:
5106:
5062:
5033:
4980:
4975:
4460:
4431:
4387:
4343:
4288:
4260:
4225:
4165:
4106:
4054:
4006:
3962:
3922:
3886:
3861:
3834:
3815:, inertia tensors
3805:
3776:
3739:
3710:
3683:
3654:
3617:
3563:
3534:
3507:
3485:using equation (5)
3475:
3455:
3426:
3397:
3368:
3339:
3310:
3283:
3256:
3227:
3196:
3162:
3115:
2807:
2772:
2680:
2651:
2571:
2506:
2486:
2446:
2362:
2298:
2276:
2228:
2178:
2129:
2085:
2057:
2004:
1934:
1881:
1809:
1760:
1716:
1688:
1592:
1493:
1463:
1434:
1402:
1373:
1338:
1278:
1246:
1217:
1171:
1124:
1090:
1037:
991:
946:
918:
894:
854:
853:
805:
778:
758:
736:
674:
662:For fixed impulse
650:
610:
583:
542:
511:
460:
414:
386:
358:
270:
237:
216:
151:
141:Rigid body contact
135:collision response
121:In the context of
6149:
6148:
6123:
6115:
6073:
6035:
6006:
5991:
5907:
5878:
5808:
5807:
5577:
5514:
5513:
5488:
5480:
5438:
5403:
5374:
5359:
5234:
5161:
5132:
5059:
4997:
4996:
4964:
4929:
4885:
4850:
4822:
4813:
4798:
4746:
4731:
4672:
4644:
4635:
4620:
4568:
4553:
4494:
4457:
4413:
4325:
4241:
4240:
3614:
3560:
3478:{\displaystyle e}
3452:
3131:
3130:
3113:
3109:
3073:
2998:
2889:
2788:
2787:
2769:
2733:
2677:
2509:{\displaystyle e}
2462:
2461:
2088:{\displaystyle i}
2073:
2072:
2051:
1928:
1719:{\displaystyle i}
1704:
1703:
1685:
1674:
1589:
1578:
1460:
1431:
1412:and its negation
1399:
1335:
824:
781:{\displaystyle 0}
240:{\displaystyle e}
116:
115:
108:
90:
6276:
6240:
6235:
6208:
6206:
6205:
6200:
6198:
6197:
6192:
6179:
6177:
6176:
6171:
6169:
6168:
6163:
6141:
6139:
6138:
6133:
6131:
6128:
6124:
6121:
6117:
6116:
6108:
6105:
6104:
6088:
6087:
6075:
6074:
6066:
6060:
6059:
6054:
6037:
6036:
6028:
6022:
6021:
6016:
6008:
6007:
5999:
5993:
5992:
5984:
5978:
5977:
5972:
5946:
5945:
5940:
5923:
5919:
5917:
5916:
5911:
5909:
5908:
5900:
5890:
5888:
5887:
5882:
5880:
5879:
5871:
5861:
5859:
5858:
5853:
5851:
5846:
5845:
5840:
5834:
5826:
5825:
5801:
5799:
5798:
5793:
5791:
5790:
5781:
5780:
5775:
5766:
5765:
5741:
5739:
5738:
5733:
5731:
5730:
5721:
5720:
5715:
5706:
5705:
5686:
5682:
5680:
5679:
5676:{\displaystyle }
5674:
5669:
5668:
5653:
5652:
5633:
5631:
5630:
5625:
5623:
5622:
5617:
5608:
5607:
5602:
5589:
5587:
5586:
5581:
5579:
5578:
5570:
5560:
5558:
5557:
5552:
5550:
5549:
5544:
5535:
5534:
5529:
5506:
5504:
5503:
5498:
5496:
5493:
5489:
5486:
5482:
5481:
5473:
5470:
5469:
5453:
5452:
5440:
5439:
5431:
5425:
5424:
5419:
5405:
5404:
5396:
5390:
5389:
5384:
5376:
5375:
5367:
5361:
5360:
5352:
5346:
5345:
5340:
5317:
5316:
5311:
5294:
5290:
5288:
5287:
5282:
5280:
5279:
5274:
5265:
5264:
5259:
5246:
5244:
5243:
5238:
5236:
5235:
5227:
5217:
5215:
5214:
5209:
5207:
5206:
5201:
5192:
5191:
5186:
5173:
5171:
5170:
5165:
5163:
5162:
5154:
5144:
5142:
5141:
5136:
5134:
5133:
5125:
5115:
5113:
5112:
5107:
5105:
5104:
5099:
5090:
5089:
5084:
5071:
5069:
5068:
5063:
5061:
5060:
5052:
5042:
5040:
5039:
5034:
5032:
5031:
5026:
5017:
5016:
5011:
4989:
4987:
4986:
4981:
4979:
4976:
4966:
4965:
4957:
4951:
4950:
4945:
4931:
4930:
4922:
4916:
4915:
4910:
4902:
4887:
4886:
4878:
4872:
4871:
4866:
4852:
4851:
4843:
4837:
4836:
4831:
4823:
4821:
4820:
4815:
4814:
4806:
4800:
4799:
4791:
4785:
4784:
4779:
4767:
4766:
4761:
4755:
4749:
4748:
4747:
4739:
4733:
4732:
4724:
4718:
4717:
4712:
4700:
4699:
4694:
4687:
4682:
4674:
4673:
4665:
4659:
4658:
4653:
4645:
4643:
4642:
4637:
4636:
4628:
4622:
4621:
4613:
4607:
4606:
4601:
4589:
4588:
4583:
4577:
4571:
4570:
4569:
4561:
4555:
4554:
4546:
4540:
4539:
4534:
4522:
4521:
4516:
4509:
4496:
4495:
4487:
4473:
4469:
4467:
4466:
4461:
4459:
4458:
4450:
4441:, orthogonal to
4440:
4438:
4437:
4432:
4430:
4429:
4424:
4415:
4414:
4406:
4396:
4394:
4393:
4388:
4386:
4385:
4380:
4371:
4370:
4365:
4352:
4350:
4349:
4344:
4342:
4341:
4336:
4327:
4326:
4318:
4297:
4295:
4294:
4289:
4287:
4286:
4269:
4267:
4266:
4261:
4259:
4258:
4234:
4232:
4231:
4226:
4224:
4223:
4214:
4213:
4208:
4199:
4198:
4174:
4172:
4171:
4166:
4164:
4163:
4154:
4153:
4148:
4139:
4138:
4119:
4115:
4113:
4112:
4107:
4105:
4100:
4099:
4094:
4088:
4080:
4079:
4063:
4061:
4060:
4055:
4053:
4045:
4044:
4032:
4031:
4015:
4013:
4012:
4007:
4005:
4004:
3999:
3990:
3989:
3984:
3971:
3969:
3968:
3963:
3961:
3953:
3952:
3947:
3934:dynamic friction
3931:
3929:
3928:
3923:
3921:
3913:
3912:
3907:
3890:Coulomb friction
3870:
3868:
3867:
3862:
3860:
3859:
3843:
3841:
3840:
3835:
3833:
3832:
3827:
3814:
3812:
3811:
3806:
3804:
3803:
3798:
3785:
3783:
3782:
3777:
3775:
3774:
3769:
3768:
3748:
3746:
3745:
3740:
3738:
3737:
3732:
3719:
3717:
3716:
3711:
3709:
3708:
3692:
3690:
3689:
3684:
3682:
3681:
3676:
3663:
3661:
3660:
3655:
3653:
3652:
3647:
3646:
3626:
3624:
3623:
3618:
3616:
3615:
3607:
3604:
3603:
3591:
3590:
3585:
3572:
3570:
3569:
3564:
3562:
3561:
3553:
3543:
3541:
3540:
3535:
3533:
3532:
3516:
3514:
3513:
3508:
3506:
3505:
3500:
3484:
3482:
3481:
3476:
3464:
3462:
3461:
3456:
3454:
3453:
3445:
3435:
3433:
3432:
3427:
3425:
3424:
3419:
3406:
3404:
3403:
3398:
3396:
3395:
3390:
3377:
3375:
3374:
3369:
3367:
3366:
3361:
3348:
3346:
3345:
3340:
3338:
3337:
3332:
3319:
3317:
3316:
3311:
3309:
3308:
3292:
3290:
3289:
3284:
3282:
3281:
3265:
3263:
3262:
3257:
3255:
3254:
3249:
3236:
3234:
3233:
3228:
3226:
3225:
3205:
3203:
3202:
3197:
3195:
3194:
3189:
3188:
3171:
3169:
3168:
3163:
3161:
3160:
3155:
3154:
3124:
3122:
3121:
3116:
3114:
3112:
3111:
3110:
3102:
3093:
3092:
3087:
3075:
3074:
3066:
3060:
3059:
3054:
3045:
3044:
3036:
3035:
3034:
3029:
3018:
3017:
3012:
3000:
2999:
2991:
2985:
2984:
2979:
2970:
2969:
2961:
2960:
2959:
2954:
2940:
2939:
2931:
2930:
2929:
2915:
2914:
2906:
2905:
2904:
2892:
2891:
2890:
2882:
2876:
2875:
2870:
2845:
2840:
2839:
2820:
2816:
2814:
2813:
2808:
2806:
2805:
2781:
2779:
2778:
2773:
2771:
2770:
2762:
2756:
2755:
2750:
2735:
2734:
2726:
2720:
2719:
2714:
2713:
2693:
2689:
2687:
2686:
2681:
2679:
2678:
2670:
2660:
2658:
2657:
2652:
2650:
2649:
2641:
2640:
2627:
2626:
2618:
2617:
2604:
2603:
2598:
2597:
2580:
2578:
2577:
2572:
2570:
2569:
2561:
2552:
2551:
2543:
2534:
2533:
2528:
2515:
2513:
2512:
2507:
2495:
2493:
2492:
2487:
2455:
2453:
2452:
2447:
2445:
2444:
2439:
2430:
2429:
2424:
2415:
2414:
2409:
2400:
2399:
2391:
2375:
2371:
2369:
2368:
2363:
2361:
2360:
2355:
2346:
2345:
2330:
2329:
2307:
2305:
2304:
2299:
2297:
2285:
2283:
2282:
2277:
2275:
2274:
2269:
2260:
2259:
2254:
2237:
2235:
2234:
2229:
2227:
2226:
2215:
2206:
2205:
2200:
2187:
2185:
2184:
2179:
2177:
2176:
2171:
2162:
2161:
2156:
2155:
2138:
2136:
2135:
2130:
2128:
2127:
2122:
2113:
2112:
2107:
2094:
2092:
2091:
2086:
2066:
2064:
2063:
2058:
2053:
2052:
2044:
2038:
2037:
2032:
2022:
2014:
2009:
2003:
2002:
1990:
1989:
1984:
1975:
1974:
1969:
1968:
1943:
1941:
1940:
1935:
1930:
1929:
1921:
1915:
1914:
1909:
1899:
1891:
1886:
1880:
1879:
1867:
1866:
1861:
1852:
1851:
1846:
1845:
1825:
1818:
1816:
1815:
1810:
1808:
1807:
1802:
1793:
1792:
1787:
1786:
1769:
1767:
1766:
1761:
1759:
1758:
1753:
1744:
1743:
1738:
1725:
1723:
1722:
1717:
1697:
1695:
1694:
1689:
1687:
1686:
1678:
1675:
1673:
1672:
1663:
1662:
1653:
1648:
1647:
1642:
1633:
1632:
1627:
1626:
1601:
1599:
1598:
1593:
1591:
1590:
1582:
1579:
1577:
1576:
1567:
1566:
1557:
1552:
1551:
1546:
1537:
1536:
1531:
1530:
1510:
1502:
1500:
1499:
1494:
1492:
1491:
1472:
1470:
1469:
1464:
1462:
1461:
1453:
1443:
1441:
1440:
1435:
1433:
1432:
1424:
1411:
1409:
1408:
1403:
1401:
1400:
1392:
1382:
1380:
1379:
1374:
1372:
1364:
1363:
1347:
1345:
1344:
1339:
1337:
1336:
1328:
1325:
1324:
1309:
1308:
1303:
1287:
1285:
1284:
1279:
1277:
1276:
1271:
1255:
1253:
1252:
1247:
1245:
1244:
1239:
1226:
1224:
1223:
1218:
1216:
1215:
1210:
1201:
1200:
1195:
1180:
1178:
1177:
1172:
1164:
1163:
1158:
1151:
1150:
1149:
1139:
1138:
1137:
1120:
1119:
1114:
1099:
1097:
1096:
1091:
1089:
1088:
1083:
1065:
1064:
1059:
1046:
1044:
1043:
1040:{\displaystyle }
1038:
1033:
1032:
1020:
1019:
1000:
998:
997:
992:
990:
989:
984:
966:
965:
960:
928:, the notion of
927:
925:
924:
919:
917:
903:
901:
900:
895:
887:
881:
880:
879:
869:
868:
867:
852:
851:
850:
838:
837:
814:
812:
811:
806:
804:
803:
787:
785:
784:
779:
767:
765:
764:
759:
757:
745:
743:
742:
737:
729:
725:
713:
712:
700:
699:
683:
681:
680:
675:
673:
659:
657:
656:
651:
643:
637:
636:
635:
625:
624:
623:
606:
592:
590:
589:
584:
582:
581:
576:
567:
551:
549:
548:
543:
532:
520:
518:
517:
512:
501:
481:
469:
467:
466:
461:
456:
455:
443:
442:
423:
421:
420:
415:
413:
395:
393:
392:
387:
385:
367:
365:
364:
359:
357:
356:
351:
333:
309:Frictional force
283:dynamic friction
246:
244:
243:
238:
225:
223:
222:
217:
129:employed within
125:simulations and
111:
104:
100:
97:
91:
89:
48:
24:
16:
6284:
6283:
6279:
6278:
6277:
6275:
6274:
6273:
6259:
6258:
6249:
6244:
6243:
6236:
6232:
6227:
6215:
6187:
6182:
6181:
6158:
6153:
6152:
6126:
6125:
6118:
6096:
6090:
6089:
6079:
6049:
6044:
6011:
6009:
5967:
5950:
5935:
5930:
5929:
5893:
5892:
5864:
5863:
5835:
5817:
5812:
5811:
5782:
5770:
5757:
5752:
5751:
5722:
5710:
5697:
5692:
5691:
5660:
5644:
5636:
5635:
5612:
5597:
5592:
5591:
5563:
5562:
5539:
5524:
5519:
5518:
5491:
5490:
5483:
5461:
5455:
5454:
5444:
5414:
5412:
5379:
5377:
5335:
5321:
5306:
5301:
5300:
5291:is computed as
5269:
5254:
5249:
5248:
5220:
5219:
5196:
5181:
5176:
5175:
5147:
5146:
5118:
5117:
5094:
5079:
5074:
5073:
5045:
5044:
5021:
5006:
5001:
5000:
4974:
4973:
4940:
4938:
4905:
4903:
4895:
4894:
4861:
4859:
4826:
4824:
4774:
4756:
4750:
4707:
4689:
4688:
4683:
4681:
4648:
4646:
4596:
4578:
4572:
4529:
4511:
4510:
4500:
4480:
4479:
4443:
4442:
4419:
4399:
4398:
4375:
4360:
4355:
4354:
4331:
4311:
4310:
4278:
4273:
4272:
4250:
4245:
4244:
4215:
4203:
4190:
4185:
4184:
4155:
4143:
4130:
4125:
4124:
4089:
4071:
4066:
4065:
4036:
4023:
4018:
4017:
3994:
3979:
3974:
3973:
3942:
3937:
3936:
3902:
3897:
3896:
3894:static friction
3878:
3851:
3846:
3845:
3822:
3817:
3816:
3793:
3788:
3787:
3761:
3759:
3754:
3753:
3727:
3722:
3721:
3700:
3695:
3694:
3671:
3666:
3665:
3639:
3637:
3632:
3631:
3595:
3580:
3575:
3574:
3546:
3545:
3524:
3519:
3518:
3495:
3490:
3489:
3467:
3466:
3438:
3437:
3414:
3409:
3408:
3385:
3380:
3379:
3356:
3351:
3350:
3327:
3322:
3321:
3300:
3295:
3294:
3273:
3268:
3267:
3244:
3239:
3238:
3217:
3212:
3211:
3206:is as follows:
3181:
3179:
3174:
3173:
3147:
3145:
3140:
3139:
3136:
3082:
3049:
3024:
3022:
3007:
2974:
2949:
2947:
2921:
2919:
2896:
2894:
2893:
2865:
2846:
2831:
2826:
2825:
2797:
2792:
2791:
2745:
2706:
2704:
2699:
2698:
2663:
2662:
2633:
2631:
2610:
2608:
2590:
2588:
2583:
2582:
2556:
2538:
2523:
2518:
2517:
2498:
2497:
2466:
2465:
2434:
2419:
2404:
2386:
2381:
2380:
2350:
2334:
2318:
2313:
2312:
2311:The velocities
2288:
2287:
2264:
2249:
2244:
2243:
2210:
2195:
2190:
2189:
2166:
2148:
2146:
2141:
2140:
2117:
2102:
2097:
2096:
2077:
2076:
2075:where, for the
2027:
1994:
1979:
1961:
1959:
1954:
1953:
1904:
1871:
1856:
1838:
1836:
1831:
1830:
1797:
1779:
1777:
1772:
1771:
1748:
1733:
1728:
1727:
1708:
1707:
1706:where, for the
1664:
1654:
1637:
1619:
1617:
1612:
1611:
1568:
1558:
1541:
1523:
1521:
1516:
1515:
1483:
1478:
1477:
1446:
1445:
1414:
1413:
1385:
1384:
1355:
1350:
1349:
1316:
1298:
1290:
1289:
1266:
1258:
1257:
1234:
1229:
1228:
1205:
1190:
1185:
1184:
1181:
1153:
1141:
1129:
1109:
1104:
1103:
1078:
1054:
1049:
1048:
1024:
1011:
1003:
1002:
979:
955:
950:
949:
938:
908:
907:
904:
871:
859:
842:
829:
819:
818:
795:
790:
789:
770:
769:
748:
747:
717:
704:
691:
686:
685:
664:
663:
660:
627:
615:
597:
596:
571:
558:
557:
556:and denoted by
523:
522:
472:
471:
447:
434:
426:
425:
398:
397:
370:
369:
346:
324:
323:
320:
311:
299:
291:
279:static friction
262:
229:
228:
196:
195:
184:
168:
143:
127:physics engines
119:
112:
101:
95:
92:
49:
47:
37:
25:
12:
11:
5:
6282:
6280:
6272:
6271:
6261:
6260:
6257:
6256:
6248:
6245:
6242:
6241:
6229:
6228:
6226:
6223:
6222:
6221:
6214:
6211:
6196:
6191:
6167:
6162:
6147:
6146:
6143:
6130:
6119:
6114:
6111:
6103:
6099:
6095:
6092:
6091:
6086:
6082:
6078:
6072:
6069:
6063:
6058:
6053:
6048:
6045:
6043:
6040:
6034:
6031:
6025:
6020:
6015:
6010:
6005:
6002:
5996:
5990:
5987:
5981:
5976:
5971:
5966:
5963:
5960:
5957:
5956:
5953:
5949:
5944:
5939:
5926:
5906:
5903:
5877:
5874:
5850:
5844:
5839:
5833:
5829:
5824:
5820:
5806:
5805:
5802:
5789:
5785:
5779:
5774:
5769:
5764:
5760:
5749:
5746:
5745:
5742:
5729:
5725:
5719:
5714:
5709:
5704:
5700:
5689:
5672:
5667:
5663:
5659:
5656:
5651:
5647:
5643:
5621:
5616:
5611:
5606:
5601:
5576:
5573:
5548:
5543:
5538:
5533:
5528:
5512:
5511:
5508:
5495:
5484:
5479:
5476:
5468:
5464:
5460:
5457:
5456:
5451:
5447:
5443:
5437:
5434:
5428:
5423:
5418:
5413:
5411:
5408:
5402:
5399:
5393:
5388:
5383:
5378:
5373:
5370:
5364:
5358:
5355:
5349:
5344:
5339:
5334:
5331:
5328:
5327:
5324:
5320:
5315:
5310:
5297:
5278:
5273:
5268:
5263:
5258:
5233:
5230:
5205:
5200:
5195:
5190:
5185:
5160:
5157:
5131:
5128:
5103:
5098:
5093:
5088:
5083:
5058:
5055:
5030:
5025:
5020:
5015:
5010:
4995:
4994:
4991:
4978:
4972:
4969:
4963:
4960:
4954:
4949:
4944:
4939:
4937:
4934:
4928:
4925:
4919:
4914:
4909:
4904:
4901:
4897:
4896:
4893:
4890:
4884:
4881:
4875:
4870:
4865:
4860:
4858:
4855:
4849:
4846:
4840:
4835:
4830:
4825:
4819:
4812:
4809:
4803:
4797:
4794:
4788:
4783:
4778:
4773:
4770:
4765:
4760:
4754:
4745:
4742:
4736:
4730:
4727:
4721:
4716:
4711:
4706:
4703:
4698:
4693:
4685:
4684:
4680:
4677:
4671:
4668:
4662:
4657:
4652:
4647:
4641:
4634:
4631:
4625:
4619:
4616:
4610:
4605:
4600:
4595:
4592:
4587:
4582:
4576:
4567:
4564:
4558:
4552:
4549:
4543:
4538:
4533:
4528:
4525:
4520:
4515:
4507:
4506:
4503:
4499:
4493:
4490:
4476:
4456:
4453:
4428:
4423:
4418:
4412:
4409:
4384:
4379:
4374:
4369:
4364:
4340:
4335:
4330:
4324:
4321:
4285:
4281:
4257:
4253:
4239:
4238:
4235:
4222:
4218:
4212:
4207:
4202:
4197:
4193:
4182:
4179:
4178:
4175:
4162:
4158:
4152:
4147:
4142:
4137:
4133:
4122:
4104:
4098:
4093:
4087:
4083:
4078:
4074:
4052:
4048:
4043:
4039:
4035:
4030:
4026:
4003:
3998:
3993:
3988:
3983:
3960:
3956:
3951:
3946:
3920:
3916:
3911:
3906:
3877:
3874:
3873:
3872:
3858:
3854:
3831:
3826:
3802:
3797:
3773:
3767:
3764:
3750:
3736:
3731:
3707:
3703:
3680:
3675:
3651:
3645:
3642:
3628:
3613:
3610:
3602:
3598:
3594:
3589:
3584:
3559:
3556:
3531:
3527:
3504:
3499:
3486:
3474:
3451:
3448:
3423:
3418:
3394:
3389:
3365:
3360:
3336:
3331:
3307:
3303:
3280:
3276:
3253:
3248:
3224:
3220:
3193:
3187:
3184:
3159:
3153:
3150:
3135:
3132:
3129:
3128:
3125:
3108:
3105:
3099:
3096:
3091:
3086:
3081:
3078:
3072:
3069:
3063:
3058:
3053:
3048:
3043:
3040:
3033:
3028:
3021:
3016:
3011:
3006:
3003:
2997:
2994:
2988:
2983:
2978:
2973:
2968:
2965:
2958:
2953:
2946:
2943:
2938:
2935:
2928:
2924:
2918:
2913:
2910:
2903:
2899:
2888:
2885:
2879:
2874:
2869:
2864:
2861:
2858:
2855:
2852:
2849:
2843:
2838:
2834:
2823:
2804:
2800:
2786:
2785:
2782:
2768:
2765:
2759:
2754:
2749:
2744:
2741:
2738:
2732:
2729:
2723:
2718:
2712:
2709:
2696:
2676:
2673:
2648:
2645:
2639:
2636:
2630:
2625:
2622:
2616:
2613:
2607:
2602:
2596:
2593:
2568:
2565:
2560:
2555:
2550:
2547:
2542:
2537:
2532:
2527:
2505:
2485:
2482:
2479:
2476:
2473:
2460:
2459:
2456:
2443:
2438:
2433:
2428:
2423:
2418:
2413:
2408:
2403:
2398:
2395:
2390:
2378:
2359:
2354:
2349:
2344:
2341:
2337:
2333:
2328:
2325:
2321:
2296:
2273:
2268:
2263:
2258:
2253:
2240:inertia tensor
2225:
2222:
2219:
2214:
2209:
2204:
2199:
2175:
2170:
2165:
2160:
2154:
2151:
2126:
2121:
2116:
2111:
2106:
2084:
2071:
2070:
2067:
2056:
2050:
2047:
2041:
2036:
2031:
2026:
2021:
2018:
2013:
2008:
2001:
1997:
1993:
1988:
1983:
1978:
1973:
1967:
1964:
1951:
1948:
1947:
1944:
1933:
1927:
1924:
1918:
1913:
1908:
1903:
1898:
1895:
1890:
1885:
1878:
1874:
1870:
1865:
1860:
1855:
1850:
1844:
1841:
1828:
1806:
1801:
1796:
1791:
1785:
1782:
1757:
1752:
1747:
1742:
1737:
1715:
1702:
1701:
1698:
1684:
1681:
1671:
1667:
1661:
1657:
1651:
1646:
1641:
1636:
1631:
1625:
1622:
1609:
1606:
1605:
1602:
1588:
1585:
1575:
1571:
1565:
1561:
1555:
1550:
1545:
1540:
1535:
1529:
1526:
1513:
1490:
1486:
1459:
1456:
1430:
1427:
1421:
1398:
1395:
1371:
1367:
1362:
1358:
1334:
1331:
1323:
1319:
1315:
1312:
1307:
1302:
1297:
1275:
1270:
1265:
1243:
1238:
1214:
1209:
1204:
1199:
1194:
1170:
1167:
1162:
1157:
1148:
1144:
1136:
1132:
1127:
1123:
1118:
1113:
1102:
1100:, computed as
1087:
1082:
1077:
1074:
1071:
1068:
1063:
1058:
1036:
1031:
1027:
1023:
1018:
1014:
1010:
988:
983:
978:
975:
972:
969:
964:
959:
937:
934:
916:
893:
890:
886:
878:
874:
866:
862:
857:
849:
845:
841:
836:
832:
827:
817:
802:
798:
777:
756:
735:
732:
728:
724:
720:
716:
711:
707:
703:
698:
694:
672:
649:
646:
642:
634:
630:
622:
618:
613:
609:
605:
595:
580:
575:
570:
566:
541:
538:
535:
531:
510:
507:
504:
500:
496:
493:
490:
487:
484:
480:
459:
454:
450:
446:
441:
437:
433:
412:
408:
405:
384:
380:
377:
355:
350:
345:
342:
339:
336:
332:
319:
316:
310:
307:
298:
295:
290:
287:
261:
258:
236:
215:
212:
209:
206:
203:
183:
180:
167:
166:Contact forces
164:
142:
139:
117:
114:
113:
28:
26:
19:
13:
10:
9:
6:
4:
3:
2:
6281:
6270:
6267:
6266:
6264:
6255:
6251:
6250:
6246:
6239:
6234:
6231:
6224:
6220:
6217:
6216:
6212:
6210:
6194:
6189:
6165:
6160:
6142:
6101:
6097:
6093:
6084:
6080:
6076:
6061:
6056:
6046:
6041:
6038:
6023:
6018:
5979:
5974:
5964:
5958:
5951:
5947:
5942:
5924:
5921:
5842:
5827:
5822:
5818:
5787:
5783:
5777:
5772:
5767:
5762:
5758:
5747:
5727:
5723:
5717:
5712:
5707:
5702:
5698:
5687:
5684:
5665:
5661:
5657:
5654:
5649:
5645:
5619:
5609:
5604:
5546:
5536:
5531:
5507:
5466:
5462:
5458:
5449:
5445:
5441:
5426:
5421:
5409:
5406:
5391:
5386:
5347:
5342:
5329:
5322:
5318:
5313:
5295:
5292:
5276:
5266:
5261:
5203:
5193:
5188:
5101:
5091:
5086:
5028:
5018:
5013:
4990:
4970:
4967:
4952:
4947:
4935:
4932:
4917:
4912:
4891:
4888:
4873:
4868:
4856:
4853:
4838:
4833:
4786:
4781:
4768:
4763:
4719:
4714:
4701:
4696:
4678:
4675:
4660:
4655:
4608:
4603:
4590:
4585:
4541:
4536:
4523:
4518:
4501:
4497:
4474:
4471:
4426:
4416:
4382:
4372:
4367:
4338:
4328:
4307:
4304:
4303:friction cone
4299:
4283:
4279:
4255:
4251:
4220:
4216:
4210:
4205:
4200:
4195:
4191:
4180:
4160:
4156:
4150:
4145:
4140:
4135:
4131:
4120:
4117:
4096:
4081:
4076:
4072:
4046:
4041:
4037:
4033:
4028:
4024:
4001:
3996:
3991:
3986:
3981:
3954:
3949:
3944:
3935:
3914:
3909:
3904:
3895:
3891:
3882:
3875:
3856:
3852:
3829:
3800:
3795:
3771:
3765:
3762:
3751:
3734:
3705:
3701:
3678:
3649:
3643:
3629:
3600:
3596:
3592:
3587:
3529:
3525:
3502:
3487:
3472:
3421:
3392:
3363:
3334:
3305:
3301:
3278:
3274:
3251:
3222:
3218:
3209:
3208:
3207:
3191:
3185:
3182:
3157:
3151:
3133:
3097:
3089:
3079:
3061:
3056:
3041:
3038:
3031:
3019:
3014:
3004:
2986:
2981:
2966:
2963:
2956:
2941:
2936:
2933:
2926:
2922:
2916:
2911:
2908:
2901:
2897:
2877:
2872:
2859:
2856:
2853:
2847:
2841:
2836:
2832:
2821:
2818:
2802:
2798:
2757:
2752:
2742:
2739:
2736:
2721:
2716:
2710:
2694:
2691:
2646:
2643:
2637:
2628:
2623:
2620:
2614:
2605:
2600:
2594:
2566:
2563:
2553:
2548:
2545:
2535:
2530:
2503:
2483:
2480:
2477:
2474:
2471:
2441:
2431:
2426:
2421:
2416:
2411:
2401:
2396:
2393:
2376:
2373:
2357:
2347:
2342:
2339:
2335:
2331:
2326:
2323:
2319:
2309:
2271:
2261:
2256:
2241:
2223:
2220:
2217:
2207:
2202:
2173:
2163:
2158:
2152:
2149:
2124:
2114:
2109:
2104:
2082:
2039:
2034:
2019:
2016:
2011:
1999:
1995:
1991:
1986:
1981:
1976:
1971:
1965:
1962:
1949:
1916:
1911:
1896:
1893:
1888:
1876:
1872:
1868:
1863:
1858:
1853:
1848:
1842:
1839:
1826:
1823:
1820:
1804:
1794:
1789:
1783:
1755:
1745:
1740:
1713:
1669:
1665:
1659:
1655:
1649:
1644:
1634:
1629:
1623:
1607:
1573:
1569:
1563:
1559:
1553:
1548:
1538:
1533:
1527:
1511:
1508:
1506:
1488:
1484:
1474:
1419:
1365:
1360:
1356:
1321:
1317:
1313:
1310:
1305:
1295:
1273:
1263:
1241:
1212:
1202:
1197:
1168:
1165:
1160:
1146:
1142:
1134:
1130:
1125:
1121:
1116:
1101:
1085:
1075:
1069:
1061:
1029:
1025:
1021:
1016:
1012:
986:
976:
970:
962:
942:
935:
933:
931:
891:
888:
876:
872:
864:
860:
855:
847:
843:
834:
830:
816:
800:
796:
775:
726:
718:
709:
705:
696:
692:
647:
644:
632:
628:
620:
616:
611:
607:
594:
578:
568:
555:
536:
505:
494:
491:
485:
452:
448:
444:
439:
435:
406:
403:
378:
375:
353:
343:
337:
317:
315:
308:
306:
304:
296:
294:
289:Applied force
288:
286:
284:
280:
274:
266:
259:
257:
255:
250:
234:
210:
204:
201:
193:
189:
181:
179:
175:
173:
165:
163:
160:
156:
147:
140:
138:
136:
132:
128:
124:
110:
107:
99:
88:
85:
81:
78:
74:
71:
67:
64:
60:
57: –
56:
52:
51:Find sources:
45:
41:
35:
34:
29:This article
27:
23:
18:
17:
6233:
6150:
5928:
5809:
5515:
5299:
4998:
4478:
4308:
4300:
4242:
3887:
3237:in terms of
3137:
2789:
2463:
2310:
2074:
1821:
1705:
1475:
1182:
947:
929:
905:
661:
321:
312:
303:Normal force
300:
297:Normal force
292:
282:
278:
275:
271:
253:
187:
185:
176:
171:
169:
155:rigid bodies
152:
134:
120:
102:
93:
83:
76:
69:
62:
50:
38:Please help
33:verification
30:
6122:(otherwise)
5487:(otherwise)
4116:as follows
2690:as follows
188:restitution
162:collision.
131:video games
6247:References
4306:friction.
4243:The value
3972:such that
66:newspapers
6269:Mechanics
6190:μ
6161:μ
6113:^
6094:−
6077:≤
6071:^
6062:⋅
6033:^
6024:⋅
6004:^
5989:^
5980:⋅
5959:−
5905:^
5876:^
5773:μ
5713:μ
5610:∈
5575:^
5537:∈
5478:^
5459:−
5442:≤
5436:^
5427:⋅
5401:^
5392:⋅
5372:^
5357:^
5348:⋅
5330:−
5267:∈
5232:^
5194:∈
5159:^
5130:^
5092:∈
5057:^
5019:∈
4962:^
4953:⋅
4927:^
4918:⋅
4889:≠
4883:^
4874:⋅
4848:^
4839:⋅
4811:^
4796:^
4787:⋅
4769:−
4744:^
4729:^
4720:⋅
4702:−
4676:≠
4670:^
4661:⋅
4633:^
4618:^
4609:⋅
4591:−
4566:^
4551:^
4542:⋅
4524:−
4492:^
4455:^
4417:∈
4411:^
4373:∈
4329:∈
4323:^
4206:μ
4146:μ
4047:∈
3997:μ
3982:μ
3955:∈
3945:μ
3915:∈
3905:μ
3796:ω
3763:ω
3693:, masses
3612:^
3558:^
3450:^
3183:ω
3107:^
3098:⋅
3080:×
3071:^
3062:×
3039:−
3005:×
2996:^
2987:×
2964:−
2934:−
2909:−
2887:^
2878:⋅
2848:−
2767:^
2758:⋅
2740:−
2731:^
2722:⋅
2675:^
2629:−
2554:−
2432:×
2422:ω
2348:∈
2262:∈
2221:×
2208:∈
2164:∈
2150:ω
2115:∈
2105:ω
2095:th body,
2049:^
2040:×
2017:−
1982:ω
1963:ω
1926:^
1917:×
1894:−
1869:−
1859:ω
1840:ω
1795:∈
1746:∈
1726:th body,
1683:^
1587:^
1554:−
1458:^
1429:^
1420:−
1397:^
1366:∈
1333:^
1314:±
1296:±
1264:−
1203:∈
1126:∫
1076:∈
977:∈
856:∫
840:→
734:∞
731:→
715:⇒
702:→
612:∫
569:∈
407:∈
379:∈
344:∈
205:∈
96:July 2017
6263:Category
6213:See also
3766:′
3644:′
3186:′
3152:′
2711:′
2638:′
2615:′
2595:′
2153:′
1966:′
1843:′
1784:′
1624:′
1528:′
521:, where
322:A force
260:Friction
182:Reaction
172:reaction
5683:yields
2817:yields
2238:is the
554:impulse
277:termed
80:scholar
5810:where
4999:where
3573:using
82:
75:
68:
61:
53:
6254:2008.
6225:Notes
6145:(10)
5804:(9b)
5744:(9a)
4237:(6b)
4177:(6a)
2069:(2b)
1946:(2a)
1700:(1b)
1604:(1a)
87:JSTOR
73:books
6180:and
5510:(8)
4993:(7)
3992:>
3932:and
3465:and
3127:(5)
2784:(4)
2464:for
2458:(3)
281:and
211:0..1
153:Two
59:news
1227:is
826:lim
42:by
6265::
3436:,
3407:,
3378:,
3349:,
3320:,
3293:,
3266:,
285:.
256:.
133:,
6195:d
6166:s
6110:t
6102:d
6098:j
6085:s
6081:j
6068:t
6057:r
6052:v
6047:m
6042:0
6039:=
6030:t
6019:r
6014:v
6001:t
5995:)
5986:t
5975:r
5970:v
5965:m
5962:(
5952:{
5948:=
5943:f
5938:j
5902:t
5873:n
5849:|
5843:r
5838:j
5832:|
5828:=
5823:r
5819:j
5788:r
5784:j
5778:d
5768:=
5763:d
5759:j
5728:r
5724:j
5718:s
5708:=
5703:s
5699:j
5671:]
5666:1
5662:t
5658:.
5655:.
5650:0
5646:t
5642:[
5620:3
5615:R
5605:r
5600:j
5572:t
5547:3
5542:R
5532:f
5527:j
5475:t
5467:d
5463:f
5450:s
5446:f
5433:t
5422:e
5417:f
5410:0
5407:=
5398:t
5387:r
5382:v
5369:t
5363:)
5354:t
5343:e
5338:f
5333:(
5323:{
5319:=
5314:f
5309:f
5277:3
5272:R
5262:f
5257:f
5229:t
5204:3
5199:R
5189:e
5184:f
5156:t
5127:n
5102:3
5097:R
5087:f
5082:f
5054:t
5029:3
5024:R
5014:e
5009:f
4971:0
4968:=
4959:n
4948:e
4943:f
4936:0
4933:=
4924:n
4913:r
4908:v
4900:0
4892:0
4880:n
4869:e
4864:f
4857:0
4854:=
4845:n
4834:r
4829:v
4818:|
4808:n
4802:)
4793:n
4782:e
4777:f
4772:(
4764:e
4759:f
4753:|
4741:n
4735:)
4726:n
4715:e
4710:f
4705:(
4697:e
4692:f
4679:0
4667:n
4656:r
4651:v
4640:|
4630:n
4624:)
4615:n
4604:r
4599:v
4594:(
4586:r
4581:v
4575:|
4563:n
4557:)
4548:n
4537:r
4532:v
4527:(
4519:r
4514:v
4502:{
4498:=
4489:t
4452:n
4427:3
4422:R
4408:t
4383:3
4378:R
4368:r
4363:v
4339:3
4334:R
4320:n
4284:d
4280:f
4256:s
4252:f
4221:r
4217:f
4211:d
4201:=
4196:d
4192:f
4161:r
4157:f
4151:s
4141:=
4136:s
4132:f
4103:|
4097:r
4092:f
4086:|
4082:=
4077:r
4073:f
4051:R
4042:d
4038:f
4034:,
4029:s
4025:f
4002:d
3987:s
3959:R
3950:d
3919:R
3910:s
3857:r
3853:j
3830:i
3825:I
3801:i
3772:i
3735:r
3730:j
3706:i
3702:m
3679:i
3674:v
3650:i
3641:v
3627:.
3609:n
3601:r
3597:j
3593:=
3588:r
3583:j
3555:n
3530:r
3526:j
3503:r
3498:j
3473:e
3447:n
3422:2
3417:r
3393:1
3388:r
3364:2
3359:I
3335:1
3330:I
3306:2
3302:m
3279:1
3275:m
3252:r
3247:v
3223:r
3219:j
3192:i
3158:i
3149:v
3104:n
3095:)
3090:2
3085:r
3077:)
3068:n
3057:2
3052:r
3047:(
3042:1
3032:2
3027:I
3020:+
3015:1
3010:r
3002:)
2993:n
2982:1
2977:r
2972:(
2967:1
2957:1
2952:I
2945:(
2942:+
2937:1
2927:2
2923:m
2917:+
2912:1
2902:1
2898:m
2884:n
2873:r
2868:v
2863:)
2860:e
2857:+
2854:1
2851:(
2842:=
2837:r
2833:j
2803:r
2799:j
2764:n
2753:r
2748:v
2743:e
2737:=
2728:n
2717:r
2708:v
2672:n
2647:1
2644:p
2635:v
2624:2
2621:p
2612:v
2606:=
2601:r
2592:v
2567:1
2564:p
2559:v
2549:2
2546:p
2541:v
2536:=
2531:r
2526:v
2504:e
2484:2
2481:,
2478:1
2475:=
2472:i
2442:i
2437:r
2427:i
2417:+
2412:i
2407:v
2402:=
2397:i
2394:p
2389:v
2358:3
2353:R
2343:2
2340:p
2336:v
2332:,
2327:1
2324:p
2320:v
2295:p
2272:3
2267:R
2257:i
2252:r
2224:3
2218:3
2213:R
2203:i
2198:I
2174:3
2169:R
2159:i
2125:3
2120:R
2110:i
2083:i
2055:)
2046:n
2035:2
2030:r
2025:(
2020:1
2012:2
2007:I
2000:r
1996:j
1992:+
1987:2
1977:=
1972:2
1932:)
1923:n
1912:1
1907:r
1902:(
1897:1
1889:1
1884:I
1877:r
1873:j
1864:1
1854:=
1849:1
1805:3
1800:R
1790:i
1781:v
1756:3
1751:R
1741:i
1736:v
1714:i
1680:n
1670:2
1666:m
1660:r
1656:j
1650:+
1645:2
1640:v
1635:=
1630:2
1621:v
1584:n
1574:1
1570:m
1564:r
1560:j
1549:1
1544:v
1539:=
1534:1
1525:v
1489:r
1485:j
1455:n
1426:n
1394:n
1370:R
1361:r
1357:j
1330:n
1322:r
1318:j
1311:=
1306:r
1301:j
1274:r
1269:j
1242:r
1237:j
1213:3
1208:R
1198:r
1193:p
1169:t
1166:d
1161:r
1156:f
1147:1
1143:t
1135:0
1131:t
1122:=
1117:r
1112:j
1086:3
1081:R
1073:)
1070:t
1067:(
1062:r
1057:j
1035:]
1030:1
1026:t
1022:,
1017:0
1013:t
1009:[
987:3
982:R
974:)
971:t
968:(
963:r
958:f
915:j
892:t
889:d
885:f
877:1
873:t
865:0
861:t
848:0
844:t
835:1
831:t
801:0
797:t
776:0
755:f
727:|
723:f
719:|
710:0
706:t
697:1
693:t
671:j
648:t
645:d
641:f
633:1
629:t
621:0
617:t
608:=
604:j
579:3
574:R
565:j
540:)
537:t
534:(
530:v
509:)
506:t
503:(
499:v
495:m
492:=
489:)
486:t
483:(
479:p
458:]
453:1
449:t
445:,
440:0
436:t
432:[
411:R
404:m
383:R
376:t
354:3
349:R
341:)
338:t
335:(
331:f
235:e
214:]
208:[
202:e
109:)
103:(
98:)
94:(
84:·
77:·
70:·
63:·
36:.
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