---
title: "Collimation: level 5 m from B on AB produced"
subject: "Surveying I"
chapter: 5
key: "semester_3/surveying_i:two-peg-50m-staff-1-875-1-790-b-5m"
source: https://www.bctengineer.com/q/semester_3/surveying_i/two-peg-50m-staff-1-875-1-790-b-5m
site: BCT Engineering
---

# Collimation: level 5 m from B on AB produced

Surveying I · Chapter 5: Leveling

- Asked in: 2072 Chaitra, 6 marks
- Video: [How to calculate the correct elevations of stations using Two Peg Test of Level|Two Peg Test Method](https://www.youtube.com/watch?v=J9P7l23-BD4) — Master Civil Engineering
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/surveying_i/two-peg-50m-staff-1-875-1-790-b-5m
- Whole chapter: https://www.bctengineer.com/md/semester_3/surveying_i/chapter-5

## Question

A levelling instrument was set up exactly midway between two pegs 50 m apart at A and B. The staff readings were 1.875 and 1.790 m respectively. The level was shifted to a point 5 m from B on the line AB produced and the staff readings at A and B were 1.630 and 1.560 m. Determine the correct staff readings when the line of collimation is exactly horizontal during the 2nd set up.

## Short answer

Collimation error -0.015 m in 50 m, precision 1:3333, line inclined downward. Correct staff readings: A = 1.6465 m, B = 1.5615 m.

## Answer

**Data:** A and B are 50 m apart. Setup 1 at the mid-point: A = 1.875 m, B = 1.790 m. Setup 2 at a point 5 m beyond B on the line AB produced (so B is 5 m and A is 55 m from the instrument): A = 1.630 m, B = 1.560 m.

#### Step 1: True difference of level (instrument at the mid-point)

With equal sight lengths the collimation error cancels, so

$$
\text{True difference (A} - \text{B)} = 1.875 - 1.790 = 0.085\ \text{m}
$$

#### Step 2: Apparent difference from the second setup

$$
\text{Apparent difference (A} - \text{B)} = 1.630 - 1.560 = 0.070\ \text{m}
$$

#### Step 3: Error and collimation precision

The reading error of a staff is proportional to its distance from the instrument. In the second setup the distances are A = 55 m and B = 5 m, so the difference of sight lengths is 50 m.

$$
e = \text{apparent} - \text{true} = 0.070 - 0.085 = -0.015\ \text{m}
$$

$$
\text{Error per metre of sight} = \frac{0.015}{50} = 0.0003\ \text{m/m}, \qquad \text{collimation precision} = \frac{1}{3333}
$$

This is 30.0 mm per 100 m of sight, which is more than the usual permissible limit of about 10 mm per 100 m (1 in 10,000) for ordinary levelling, so the instrument needs adjustment.

The readings are too small and the error grows with distance (the far staff reads too little), so the line of collimation is **inclined downward**.

#### Step 4: Correct readings for a horizontal line of sight (second setup)

Error in reading on A $= -0.0003 \times 55 = -0.0165$ m; error in reading on B $= -0.0003 \times 5 = -0.0015$ m.

$$
A = 1.630 - (-0.0165) = 1.6465\ \text{m}, \qquad B = 1.560 - (-0.0015) = 1.5615\ \text{m}
$$

Check: A $-$ B $= 1.6465 - 1.5615 = 0.085$ m, equal to the true difference.

**Answer:** with a horizontal line of sight the readings in the second setup are **A = 1.6465 m and B = 1.5615 m**; the line of collimation is inclined downward and the precision is 1 in 3333 (30 mm per 100 m), which needs adjustment.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. True difference A - B from the mid-point setup is:
   A. 0.085 m
   B. 0.070 m
   C. 0.015 m
   D. 3.665 m
   Answer: A. 1.875 - 1.790 = 0.085 m. (Key point: True difference)

2. Instrument is 5 m beyond B on AB produced. Distances are:
   A. A = 5, B = 55
   B. A = 55 m, B = 5 m
   C. A = 50, B = 5
   D. A = 45, B = 5
   Answer: B. Sight difference 50 m. (Key point: Distances)

3. Apparent 0.070 minus true 0.085 gives:
   A. -0.155 m
   B. 0.085 m
   C. -0.015 m, so downward
   D. +0.015 m, so upward
   Answer: C. Far staff A reads too little, the line points downward. (Key point: Error)

4. Corrected readings are:
   A. A = 1.6135, B = 1.5585
   B. A = 1.630, B = 1.560
   C. A = 1.6465, B = 1.5585
   D. A = 1.6465, B = 1.5615
   Answer: D. Add 0.0165 and 0.0015. (Key point: Corrected readings)

## Related in Surveying I

- [Collimation: pegs 60 m apart, Q near B](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Atwo-peg-pegs-ab-60m-q-6m-from-b) — Surveying I, chapter 5
- [Level book, staff readings at B and C (1251.250)](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Alevel-book-15m-b-c-rl-1250-800-1251-250) — Surveying I, chapter 5, asked 3×
- [Level book, staff readings at B and C (1251.000)](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Alevel-book-15m-b-c-rl-1250-800-1251-000) — Surveying I, chapter 5, asked 3×
- [Fly levelling, BM 1470 m; set 4 pegs](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Afly-level-set-4-pegs-1470-rising-1-in-200) — Surveying I, chapter 5
- [Level book: profile levelling readings](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Aprofile-leveling-readings-3rd-7th-9th) — Surveying I, chapter 5
- [Two peg test: collimation inclined up or down](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Atwo-peg-dumpy-cd-df-120m) — Surveying I, chapter 5