Question

The angle of banking $\theta$ for a meter gauge railway line is given by $\theta = \tan^{-1}\left(\frac{1}{20}\right)$. What is the elevation of the outer rail above the inner rail?

• A. $20\text{ cm}$
• B. $10\text{ cm}$
• C. $0.2\text{ cm}$
• D. $5\text{ cm}$

Hint:

Always remember the technical definitions for track gauges in physics word problems:
Meter Gauge $= 1\text{ meter} = 100\text{ cm}$.
Knowing this definition provides the missing numerical value for the width ($w$) needed to solve the problem.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
The problem describes a banked railway track configuration. We are given the banking angle parameter $\theta$ and need to find the vertical height elevation ($h$) of the outer rail relative to the inner rail. The track is specified as a "meter gauge" line, which means the width distance separating the two rails is exactly $1\text{ meter}$.

Step 2: Key Formula or Approach:
From the geometry of a banked track cross-section, the rails form a right-angled triangle where: The hypotenuse represents the track width distance ($w = 1\text{ m}$). The side opposite to the angle $\theta$ represents the vertical elevation height ($h$). For small banking angles, the tangent of the angle can be approximated using basic trigonometry: $$\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}} \approx \frac{h}{w}$$ Alternatively, using the strict definition $\sin \theta = \frac{h}{w}$, since $\tan \theta = \frac{1}{20}$ is very small, we can approximate $\sin \theta \approx \tan \theta = \frac{1}{20}$.

Step 3: Detailed Explanation:
Identify the parameters from the problem statement: Gauge width of the railway line, $w = 1\text{ m} = 100\text{ cm}$ Banking angle relation, $\tan \theta = \frac{1}{20}$ Set up the geometric relationship for the vertical elevation: $$\tan \theta = \frac{h}{w} \implies \frac{1}{20} = \frac{h}{100\text{ cm}}$$ Solve for the unknown height parameter $h$ by cross-multiplying: $$h = \frac{100\text{ cm}}{20} = 5\text{ cm}$$ The outer rail must be raised by exactly $5\text{ cm}$ above the level of the inner rail.

Step 4: Final Answer:
The elevation height of the outer rail is $5\text{ cm}$, matching option (D).