TIFR CSE 2019 | Part A | Question: 8

Question

Consider the following toy model of traffic on a straight , single lane, highway. We think of cars as points, which move at the maximum speed $v$ that satisfies the following constraints:

1. The speed is no more than the speed limit $v_{max}$ mandated for the highway.
2. The speed is such that when traveling at this speed, it takes at least time $t_0$ (where  $t_0$ is a fixed time representing the reaction time of drivers) to reach the car ahead, in case the car ahead stops suddenly.

Let as assume that in the steady state, all cars on the highway move at the same speed $v$ satisfying both the above constraints, and the distance between any two successive cars is the same. Let $\rho$ denote the “density” , that is, the number of card per unit length of the highway. Which of the following graphs most accurately captures the relationship between the speed $v$ and the density $\rho$ in this model ?

• A.     
• B.       
• C.      
• D.     
• E.   

Comments

Arjun sir as density increases velocity of all cars decreases linearly. 

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C seems logical. Hope it will be the correct one.

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I think it will be A

Answer 1

As cars are moving at constant speed there is no chance of collision until all cars occupy $v*t_0$ space. So upto certain limit which is $v*t_0$ there is no effect of density of speed. After that, the system seems to follow mass flow rate equation which implies $v \propto\frac{1}{\rho}$. So, C is correct.