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For part (a) you need to observe that the flow rate \(f(x,t)\) in vehicles per unit time is \(u(x,t) \rho(x,t)\).
The next step is to write down the partial differential equation satisfied by the traffic density. This is derivable from a conservation of mass (or vehicle numbers) argument that you will have seen innumerable times.
Thank you very much captainBlack. im very humble with your reply but same time i m lost with your answers provided. Would please go through little deeply and clarify it nicely please. ThnxThe next step is to write down the partial differential equation satisfied by the traffic density. This is derivable from a conservation of mass (or vehicle numbers) argument that you will have seen innumerable times.
CB
Consider a road element between \(x\) and \(x+\Delta x\) the traffic flow into the element at \(x\) per unit time is \(u(\rho(x,t))\rho(x,t)\) and out at \(x+\Delta x\) is \(u(\rho(x+\Delta x,t))\rho(x+\Delta x,t)\) Therefore the rate of change of car numbers in the element is:Thank you very much captainBlack. im very humble with your reply but same time i m lost with your answers provided. Would please go through little deeply and clarify it nicely please. Thnx
In small time interval \(\Delta t\) all the vehicles less than a distance \(u(x,t)\Delta t\) down stream of \(x\) will pass \(x\). The number of vehicles in this stretch of road is \(u(x,t)\rho(x,t)\Delta t\), so \(u(x,t)\rho(x,t)\Delta t\) vehicles pass \(x\) in \(\Delta t\) so the vehicle flow rate at \(x\) is \(u(x,t)\rho(x,t)\) vehicles per unit time.For part (a) you need to observe that the flow rate \(f(x,t)\) in vehicles per unit time is \(u(x,t) \rho(x,t)\).
Now you need to show that for (i) and (ii) that \(u(x,t)\le u_{sl}\), then as \(\rho(x,t) \le \rho_{max}\) you will have shown that the flow rate:
\[f(x,t)\le u_{sl}\rho_{max}\]