Turnout geometry: If α is the switch angle and R is the radius of the turnout curve, what is the approximate length of the tongue rail?

Introduction / Context:
In the design of railway turnouts (switches), the tongue rail must develop the required lateral divergence smoothly. For small switch angles, simple geometric relations give an estimate of the tongue length in terms of turnout radius R and switch angle α.

Given Data / Assumptions:

• Turnout curve approximated by a circular arc of radius R.
• Small-angle trigonometry is applicable for α (typical in practice).
• Length sought is the effective length of tongue from toe to where full switch angle is achieved.

Concept / Approach:
Relating the lateral set (offset) to the developed angle in a circular arc gives the approximation: length along the tongue required to generate angle α is proportional to tan(α/2) for small angles when constructing the geometric triangle of divergence.

Step-by-Step Solution:

Verification / Alternative check:
Turnout handbooks commonly use R tan(α/2) as a first-order estimate; more exact layouts refine with actual switch geometry and offsets.

Why Other Options Are Wrong:

• R tan α: Overestimates length for small angles.
• R sin α or R sin(α/2): Not the standard relation tied to the lateral divergence triangle for switch layout.
• R cos(α/2): Unrelated to length development here.

Common Pitfalls:

• Applying large-angle trigonometry without small-angle context.
• Confusing lead length of turnout with tongue length.

Final Answer:
R tan(α/2)