To derive an expression for the car's maximum speed within the time interval $0 \leq t \leq \frac{3\pi}{2q}$, we need to first define what speed refers to in this context.

Step-by-Step Process:

1. Speed and Velocity: Speed is the magnitude of the velocity function. If we have a position function $x(t)$, the velocity $v(t)$ is the first derivative of the position function with respect to time: $v(t) = \frac{dx(t)}{dt}$ The speed is the absolute value of velocity: $\text{Speed} = |v(t)|$

2. Form of the Position Function: Assume the position function $x(t)$ is periodic, because the interval involves $\pi$ and $q$, which often implies trigonometric functions like sine or cosine. A common form for such a position function is: $x(t) = A \sin(qt)$ where $A$ is the amplitude and $q$ is related to the frequency.

3. Find the Velocity Function: The velocity is the derivative of $x(t)$: $v(t) = \frac{d}{dt} [A \sin(qt)] = A q \cos(qt)$

4. Find the Critical Points: To find when the speed is maximum, we examine the velocity function and set its derivative equal to zero. The derivative of velocity is: $a(t) = \frac{dv(t)}{dt} = -A q^2 \sin(qt)$ Setting $a(t) = 0$, we get