$26-1.$ Two signals are spaced at $1,000$ feet on en urban arterial. It is desired to establish the offset between these two signals, considering only the primary flow in one direction. The desired progression speed is $40mph.$ The cycle length is $60$ seconds. Saturation headway may be taken as $2\mathrm{.}0s\frac{ec}{\text{v}\text{e}\text{h}},$ and the startup lost time as $2\mathrm{.}0$ seconds.

$($a$)$ What is the ideal offset between the two intersections assuming that vehicles arriving at the upstream intersection are already in a progression $($i$.$e$.,$ a moving platoon$)$ at the initiation of the green?

$($b$)$ What is the ideal offset between the two intersections assuming that the upstream signal is the first in the progression $($i$.$e$.,$ vehicles are starting from a standing queue$)?$

$($c$)$ What is the ideal offset assuming that an average queue of three vehicles per lane is expected at the downstream intersection at the initiation of the green? Assume the base conditions of part a$.$

$($d$)$ Consider the offset of part a$.$ What is the resulting offset in the opposite $($off$-$peak$)$ direction? What impact will this have on traffic traveling in the opposite direction?

$($e$)$ Consider the offset of part a$.$ If the progression speed was improperly estimated, and the actual desired speed of drivers was $45m\frac{i}{h},$ what impact would this have on the primary direction progression?