Turnout geometry: If L1 and L2 are the actual and theoretical lengths of a tongue rail, d is the heel divergence, and t is the tongue thickness at the toe, what is the expression for the switch angle α?

Introduction / Context:
The geometry of a switch (points) in a turnout depends on the switch angle α, which relates the lateral divergence achieved to the longitudinal development of the tongue rail. Practical formulas account for the finite thickness of the tongue at the toe and the heel divergence relative to the stock rail.

Given Data / Assumptions:

• L1: Actual tongue length; L2: Theoretical tongue length to the virtual point.
• d: Heel divergence (lateral offset at heel).
• t: Tongue thickness at the toe.
• Small-angle approximation tan α ≈ α in radians is often used, but we keep tan α explicitly.

Concept / Approach:
Lateral clearance available for the gauge face is effectively (d - t), since t occupies part of the gap near the toe. The effective run available to develop this clearance is (L1 - L2) considering actual vs. theoretical length. Hence tan α is the ratio of effective divergence to effective run.

Step-by-Step Solution:

Verification / Alternative check:
This relation is consistent with standard turnout design derivations in which the toe thickness reduces useful clearance at the onset of switch movement, adjusting the simple d/(length) triangle to (d - t).

Why Other Options Are Wrong:

• Option B: Uses sums, not differences; does not reflect geometric deduction.
• Option C: Ignores correction for L2, overestimating the angle.
• Option D: Ignores tongue thickness t; yields an optimistic (larger) angle.
• Option E: Uses only thickness, missing divergence term.

Common Pitfalls:

• Neglecting toe thickness when computing switch geometry.
• Mistaking actual vs. theoretical tongue length definitions.

Final Answer:
tan α = (d - t) / (L1 - L2)