Question

In the figure below, \( I_1 \) and \( I_2 \) are the two instrument stations. The instrument stations and the object \( P \) lie in the same vertical plane. Assume all instruments and staff are levelled.

\( L \) = horizontal distance between the object and the station \( I_2 \)
\( b \) = horizontal distance between the instrument stations
\( S \) = staff reading at the Benchmark (BM) for horizontal line of sight
\( H \) = reading on the staff at \( P \)
\( r \) = height of the point sighted by instruments at the staff kept at \( P \) from the line of sight of the instruments
\( \alpha_1 \) and \( \alpha_2 \) = vertical angles to the reading on staff at \( P \) from \( I_1 \) and \( I_2 \), respectively

Which one of the following relationships is correct for \( L \)?

 

• A. \( L = \left( \dfrac{b \tan \alpha_1}{\tan \alpha_1 - \tan \alpha_2} \right) \)
• B. \( L = \left( \dfrac{b \tan \alpha_2}{\tan \alpha_2 - \tan \alpha_1} \right) \)
• C. \( L = \left( \dfrac{b \tan \alpha_1}{\tan \alpha_2 - \tan \alpha_1} \right) \)
• D. \( L = \left( \dfrac{b \tan \alpha_2}{\tan \alpha_1 - \tan \alpha_2} \right) \)

Hint:

In two-peg test or reciprocal levelling, when both instruments observe the same elevated point from different stations, equating the vertical projections gives a solvable expression for horizontal distance.

Answer 1

The Correct Option is C

From the geometry of reciprocal levelling in a vertical plane:

- Let \( h_1 = L \tan \alpha_2 \): height of staff reading at \( P \) from \( I_2 \)
- Let \( h_2 = (b + L) \tan \alpha_1 \): height of staff reading at \( P \) from \( I_1 \)

Since both are sighting the same point at height \( r \), equate the expressions:

\[ (b + L) \tan \alpha_1 = L \tan \alpha_2 \]

Rearranging:

\[ b \tan \alpha_1 = L(\tan \alpha_2 - \tan \alpha_1) \]

Solving for \( L \):

\[ L = \frac{b \tan \alpha_1}{\tan \alpha_2 - \tan \alpha_1} \]

\[ \boxed{{Hence, option (C) is the correct relation.}} \]