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The image contains a problem related to the "car following model" governed by a system of differential equations. Here's a summary of the problem:

### The System of Differential Equations:
1. \( \dot{h}_i = v_{i+1} - v_i \)
2. \( \dot{v}_i = V(h_i) - v_i \)

Where:
- \( h_i \) is the headway (distance between cars).
- \( v_i \) is the velocity of car \( i \).
- \( V(h_i) \) is the optimal velocity function, which is piecewise defined as follows:
  \[
  V(h) = 
  \begin{cases} 
    0 & 0 \leq h \leq h_{\text{min}} \\
    s(h - h_{\text{min}}) & h_{\text{min}} < h \leq \frac{V_t}{s} + h_{\text{min}} \\
    V_t & \frac{V_t}{s} + h_{\text{min}} < h
  \end{cases}
  \]
- \( h_{\text{min}} \) is the minimum headway for stopping.
- \( V_t \) is the target velocity (speed limit).
- \( s > 0 \) is a real parameter.

### Questions:
1. **(a)** (5 marks) Sketch \( V(h) \) for different values of \( s > 0 \) to determine the influence of the parameter \( s \).
   
2. **(b)** (5 marks) Show that \( h_i(t) = h^* \), \( v_i(t) = V(h^*) \) is a steady-state solution of the system for any given \( h^* > 0 \).

Would you like a solution for part (a) or (b), or both? Feel free to ask for further clarifications or details!

Consider the car following model given by the system of differential equations for headway hi and velocity vi of car i, where the sensitivity is set to 1. The optimal velocity function V(h) is piecewise defined. (a) Sketch V(h) for different values of s > 0. (b) Show that hi(t) = h* and vi(t) = V(h*) is a steady-state solution.

Explore the car following model governed by a system of differential equations. The problem involves sketching the optimal velocity function V(h) and proving a steady-state solution for headway and velocity. Key topics include differential equations, piecewise functions, and stability analysis in traffic flow.

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