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: \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!