$($a$)$ A rapid transit system employs vehicles that can be connected in transit units.

Calculate the capacity $($vehicles per hour$)$ and the speed at capacity $($m$/$s$)$ by varying the number of

vehicles per train $(N)$ from $1$ to $5.$ Assume a perception$-$reaction time of $1\mathrm{.}5$ seconds, a vehicle length of $12$

meters, a normal deceleration rate of $1\mathrm{.}5\frac{m}{s^2}$ and a clearance length $x_0=1\mathrm{.}2$ meters. Use safety regime

$"$a$",$ that is deceleration of leading vehicle corresponding to an instantaneous stop and that of the following

vehicle to a normal stop.

$s=V*+\frac{V^2}{2d_f}-\frac{v^2}{2d_l}+N*L+x_0$

$($b$)$ Sketch the relationship between vehicles per hour and number of vehicles per train.