A freight train consists of a single $150\text{-tonne}$ locomotive hauling fifty $80\text{-tonne}$ loaded freight cars. The train is traveling at a constant speed of $80 \text{ km/h}$ on straight, level track. Using a simplified Davis equation format $R = (A + BV + CV^2) \times W$ (where $W$ is the total weight in tonnes), calculate the total tractive resistance. Assume the coefficients for this specific train configuration are:

• $A = 1.3$ (Journal friction)
• $B = 0.04$ (Flange friction)
• $C = 0.0005$ (Air resistance factor per tonne)

A locomotive has a maximum available Tractive Effort ($TE$) of $45,000 \text{ kgf}$. It is hauling the same train from the previous problem ($W = 4,150 \text{ tonnes}$) at a slow, steady speed where the base rolling resistance is $4.0 \text{ kgf/tonne}$. The train approaches a steep grade of $1.5\%$ that also features a sharp $3\text{-degree}$ horizontal curve. Calculate the total resistance on this section and determine if the locomotive has enough power to pull the train up the hill without stalling.

A standard right-hand turnout diverges from a straight main track. The number of the crossing (often called the frog number, $N$) is $12$. The track gauge ($G$) is standard gauge at $1.435 \text{ meters}$. Calculate the theoretical crossing angle ($\alpha$) in degrees and estimate the curve lead (the distance from the theoretical point of the switch to the theoretical point of the crossing) using the Right Angle method approximation ($Lead \approx 2GN$).

Trace the evolution of train control systems by explaining the operational limitations of the "Absolute Block" system using manual signals, and how modern Positive Train Control (PTC) solves these limitations to prevent collisions.