. (50 marks) Consider the car following model given by the system of differential equations
h˙
i = vi+1 − vi
, v˙i = (V (hi) − vi),
for headway hi and velocity vi of car i (where the sensitivity α is set to 1). Here the optimal velocity
function V (h) is given by
V (h) =



0 0 ≤ h ≤ hmin ,
s(h − hmin) hmin < h ≤
Vt
s
+ hmin ,
Vt
Vt
s
+ hmin < h < ∞ ,
Maths 765, Semester 2 of 2024 Assignment 2 Page 1 of 2
where hmin is the minimum headway for stopping, Vt
is the target velocity (speed limit), and s > 0
is a real parameter.
(a) (5 marks) Sketch V (h) for different values of s > 0 to determine the
influence of the parameter s.


### Problem Breakdown

The problem involves understanding the optimal velocity function \( V(h) \), which is piecewise-defined and describes how the velocity of a car \( v_i \) depends on the headway \( h_i \) (the distance to the next car ahead). The function also includes parameters like \( h_{\text{min}} \) (the minimum headway for stopping), \( V_t \) (the target velocity or speed limit), and \( s \) (a parameter that influences the slope of the velocity function). Our goal is to sketch \( V(h) \) for different values of \( s > 0 \) and analyze its behavior.

### Optimal Velocity Function \( V(h) \)

The piecewise function \( V(h) \) is defined as follows:

\[
V(h) = 
\begin{cases} 
0 & \text{for } 0 \leq h \leq h_{\text{min}}, \\
s(h - h_{\text{min}}) & \text{for } h_{\text{min}} < h \leq \frac{V_t}{s} + h_{\text{min}}, \\
V_t & \text{for } h > \frac{V_t}{s} + h_{\text{min}}.
\end{cases}
\]

This function has three parts:
1. **For headway \( h \leq h_{\text{min}} \)**: The velocity \( V(h) = 0 \). This reflects that if the distance to the next car is too small (less than or equal to the minimum headway), the car should stop.
2. **For headway \( h_{\text{min}} < h \leq \frac{V_t}{s} + h_{\text{min}} \)**: The velocity increases linearly with the slope \( s \) as \( V(h) = s(h - h_{\text{min}}) \). Here, the parameter \( s \) controls how fast the velocity increases as the headway grows.
3. **For headway \( h > \frac{V_t}{s} + h_{\text{min}} \)**: The velocity reaches a constant value \( V_t \), which represents the speed limit.

### Effect of \( s \) on \( V(h) \)

- **If \( s \) is large**, the slope in the linear region is steeper, meaning the car's velocity increases rapidly as the headway grows.
- **If \( s \) is small**, the slope in the linear region is shallower, meaning the car’s velocity increases more gradually with increasing headway.

### Sketching \( V(h) \)

1. For small \( s \), the graph has a slower rate of increase in the middle region.
2. For large \( s \), the increase in velocity is rapid in the middle region, approaching the target velocity \( V_t \) more quickly.

The key regions in the sketch:
- A flat line at \( V(h) = 0 \) for \( h \leq h_{\text{min}} \).
- A straight line with slope \( s \) for \( h_{\text{min}} < h \leq \frac{V_t}{s} + h_{\text{min}} \).
- A flat line at \( V(h) = V_t \) for \( h > \frac{V_t}{s} + h_{\text{min}} \).

### Sketch Explanation
For \( s > 0 \), you would get a family of curves, each with a steeper or shallower slope in the middle region, depending on the value of \( s \). The larger the value of \( s \), the faster the transition from \( V = 0 \) to \( V = V_t \).

---

Would you like more details on how to sketch this, or need any clarifications?

### Related Questions
1. What would happen to the velocity function if \( s = 0 \)?
2. How does the headway distance affect the stability of traffic flow based on this model?
3. How does the function change if \( V_t \) is lowered (for example, speed limits are reduced)?
4. What is the physical interpretation of the parameter \( h_{\text{min}} \)?
5. How would the model change if the target velocity \( V_t \) were a function of time?

### Tip
Understanding how parameters influence a system in piecewise functions like this is key in analyzing dynamic systems like traffic flow models.

Consider the car following model given by the system of differential equations: h˙i = vi+1 − vi , v˙i = (V(hi) − vi), for headway hi and velocity vi of car i (where the sensitivity α is set to 1). Here the optimal velocity function V(h) is given by a piecewise-defined function involving parameters hmin, Vt (target velocity), and s (a real parameter). The problem asks to sketch V(h) for different values of s > 0 to analyze its influence.

Explore the car following model with a system of differential equations that describes traffic flow. Learn how the optimal velocity function V(h) changes with different values of s, affecting headway and vehicle velocity. This problem covers advanced topics in differential equations and traffic flow dynamics, suitable for undergraduate students.

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