Trafic jam SLIDE 4 Consider PR=Pmax and PL Pmax = Shock Models moving cars encountering a traffic jam and instantaneously stopping VEHICLE TRAJECTORIEs C CHAHACTERISTCS 8<0 and the shock propagates to the left 山 Note 11 Trafic flow shock solution The cars are moving at speed ul >0 when they encounter the traffic jam. When they slam on their brakes, they instant aneously reduce their velocity to zero, and the density jumps from pr to Pmax (this is the shock). The shock location moves to the left as more cars join the traffic jam(s< 0). We can verify that characteristics run into the shock and that the entropy condition is thus satisfied 512“ Green Light Consider 0< PR< PL Pmax Rarefaction This might model the startup of cars after a light turns green VEHICLE TRAJECTOR ES需 HARACTERISICS Note 11 Trafic rarefaction solution The cars to the left initially have zero velocity(they are stopped at the inter section). They begin to accelerate as the cars in of them move. Si the velocity is inversely proportional to the density, each car can only speed upÑ✁Ò❤Ó✶Ò❤Ó Ô☛Õ❯Ö◗×➯Ø✄Ù✰Ú✁×PÛ✄Ü Ý✶Þ✧ß♠à✸á♣â❳ã ä❯å☛æ❳ç■è❭é➯ê⑧ë✏ì✣íïîðì✌ñ➉ò⑧ó➇ô☛æ❱é➇ì✧õ♣ö✫ì✧ñ➉ò⑧óø÷ ù✸ú✼û Ù✥ü ý♣þ✥ÿ✁￾✄✂✆☎✞✝✞þ✠✟☛✡✌☞✎✍✑✏✓✒✕✔✓☎✖￾✗☞✘✏➘þ✕✙✚☞✜✛✢￾✗✔✣✡✌☞✤✍✥✒✦✛✧✔★✒✪✩✫✏✭✬✗✒✕✝✮✒✕☞➭ÿ✦✡✌☞✯☎✣✛✰✒✕☞✜✛✢✒✠☞✘￾➘þ✕✙✎☎✣✂✆✱✲☎✣✛þ✴✳✵✳✶✡✌☞✤✍✤✷ VEHICLE TRAJECTORIES CHARACTERISTICS ✸ ö✫✹➛ô✒æ❱é✻✺★✼❱ê➉ç✪✼På✚✽✓✾❀✿❱ë✛å✁✿❱ô✠❁✧ô✕✺✛ê◆ç❂✺✙å✲✺★✼Pê❄❃❚ê✄❅✌✺ ❆✞❇✁❇ ❈❊❉●❋✣❍❏■✘■ ❑▼▲✕◆✎❖◗P✻❘❚❙✰❉❱❯❳❲❩❨❭❬✭❉❪P✣❫❴❨✵❉✎❙✴❵❛❋★❜✰❉✚❝ ❞✼Pê❡✽✥ô✒ë➘ç➛ô✒ë✙ê✑❢✞å✕❣✣ètæ❱❁ïô✠✺❁ç✪✿✶ê⑧ê◆é✫❤✶õ❥✐❦✹♠❧❚✼Pê✥æ✫✺★✼Pê☛♥✭ê⑧æ✜✽❼å✵♦❱æ✁✺✙ê⑧ë✲✺✴✼Pê♣✺✛ë➘ô✕q❀✽sr■ô✠❢♣t ✉✼Pê✥æ✈✺★✼Pê☛♥❸ç✪❃❭ô✠❢ å☛æ♣✺✴✼Pê⑧è❚ë❩✇Pë✙ô✵✾☛ê◆ç✗①❱✺★✼Pê☛♥➇è❚æ❱ç✪✺✙ô☛æ✁✺➘ô✒æPê✥å✵♦❱ç★❃②♥❁ë✙ê✥é✚♦❱✽⑧ê❄✺★✼❱ê⑧è❚ë③❣✌ê✗❃❚å✎✽⑧è④✺⑤♥✑✺✛å ⑥ê✥ë✛å✜①✼ô☛æ❱é❊✺★✼Pê➇é➯ê⑧æ❳ç■è⑦✺⑤♥✻r⑤♦❱❢s✿❳ç✞❅✖ë✙å✵❢❷ìõ ✺✛å♣ìñ➉ò⑧ó⑨⑧ ✺★✼❱è❚ç➪è❚ç✞✺★✼Pê➇ç✪✼På✚✽✓✾✯⑩✣t ❞✼Pê❅ç✪✼På✚✽✓✾ ❃❚å✎✽✥ô✕✺✙ètå✌æ❶❢✠å✕❣✌ê✥ç❷✺✙å❡✺✴✼Pê✻❃❚ê✄❅✌✺✞ô☛ç✞❢✞å✌ë✛ê✻✽✥ô✒ë➘ç❸r★å☛è❚æ❶✺✴✼Pê❀✺✙ë✙ô✠q✦✽❄r■ô✠❢ ⑧ ✸ ö❹✹✁⑩✄t ✉ê✥✽⑧ô☛æ ❣☛ê✥ë✛è⑦❅❺♥✦✺✴✼❱ô✕✺③✽✓✼❳ô✒ë➘ô✵✽✣✺✙ê⑧ë✙è❚ç✪✺✛è❻✽⑧ç✏ë✴♦Pæ✰è❚æ✁✺✙å✲✺★✼Pê❑ç✪✼På✚✽✓✾➇ô✒æ❱é✑✺★✼❳ô✕✺❼✺✴✼Pê➉ê⑧æ✤✺✙ë✛å✁✿✤♥✑✽❼å☛æ❳é➯è④✺✙ètå✌æ✰è❚ç ✺★✼✎♦❱ç❾ç✙ô✕✺✙è❚ç✪❽❱ê◆é●t Ñ✁Ò❤Ó✶Ò❻❾ Ô✤❿❅Ö✕➀✎➀✎➁➃➂✭➄➆➅ú✶➇ Ü Ý✶Þ✧ß♠à✸á♣âPâ ä❯å☛æ❳ç■è❭é➯ê⑧ë❚✹✞ö ì✧í✭ö✫ì✣õ❃ö ì✧ñ➉ò❼ó❷÷ ➈×➯Ö✕➀✎➉×PÙ➇ ➄û➁t ➊✶➋✎✡❻☎✞✝s✡④✍❭➋✚✛➌✝➛þ◆ÿ✵￾✗✂❛✛❺➋✯￾❷☎✣✛✰✒✕✔✣✛✧✙❭✳❞þ✪➍✞✏✴✒✠✔✓☎✖✒⑤➍✓✛✰￾✄✔➎✒❀✂✆✡④✍➏➋✎✛➐✛➑✙✚✔✣☞❱☎❚✍✵✔★￾✓￾✄☞✶✷ VEHICLE TRAJECTORIES CHARACTERISTICS ìõ î➆ìñ➉ò⑧ó ①❅ìí î➓➒ ➔ ìñ➉ò⑧ó ❆✞❇➏→ ❈❊❉●❋✣❍❏■✘■ ❑▼▲✕◆✎❖◗P❀➣➌❉❱❯s❲❄▲✕◆✚▲✁❍✵↔✢◆✘P✤❋★❜✰❉✚❝⑨❨✵❉✎❙✴❵❛❋★❜✰❉✚❝ ❞✼Pês✽⑧ô✒ë➘ç↕✺✛å❀✺✴✼Pês❃tê✗❅✌✺✳è❚æPè⑦✺✛è❭ô✠❃②❃⑦♥✈✼❱ô➏❣✌ê ⑥ ê⑧ë✙å❀❣☛ê☛❃tå✚✽❼è⑦✺⑤♥ ⑧ ✺✴✼Pê✗♥❃ô☛ë✛ê➪ç✪✺✛å✁✿✯✿✶ê✥é♣ô✠✺③✺✴✼Pê✠ètæ✤✺✛ê✥ë✪➙ ç✛ê☛✽✣✺✙ètå✌æ✜⑩✣t ❞✼Pê☛♥❶✇❳ê☛❁☛è❚æ➛✺✙å➡ô✁✽✗✽❼ê☛❃tê✥ë✙ô✠✺✛ê❁ô✌ç✞✺✴✼Pê✈✽⑧ô☛ë✙ç➪ètæ❴❅✖ë✛å✌æ✤✺❑å✠❅↕✺★✼Pê☛❢➜❢✠å✕❣✌ê✵t➛➝✣è❚æ❱✽❼ê ✺★✼❱ê③❣✌ê✗❃❚å✎✽⑧è④✺⑤♥➛è❭ç❍ètæ✎❣✌ê⑧ë➘ç■ê☛❃⑦♥✲✿Pë✙å✵✿✶å☛ë★✺✛è❚å☛æ❱ô✵❃✯✺✙å✖✺✴✼Pê❲é➯ê⑧æ❱ç✛è⑦✺⑤♥✵①➯ê✥ô✁✽✓✼✻✽⑧ô☛ë➐✽✥ô✒æ❅å☛æ✯❃②♥✔ç✪✿✶ê⑧ê◆é✻♦❱✿ ❇☛➞