---
title: "Collimation error from 5 m and 45 m setup"
subject: "Surveying I"
chapter: 5
key: "semester_3/surveying_i:two-peg-collimation-ab-50m-true-diff-0-250"
source: https://www.bctengineer.com/q/semester_3/surveying_i/two-peg-collimation-ab-50m-true-diff-0-250
site: BCT Engineering
---

# Collimation error from 5 m and 45 m setup

Surveying I · Chapter 5: Leveling

- Asked in: 2080 Bhadra, 5 marks
- Video: [Finding Collimation Error of a Level using Two Peg Test Method|Solved Problem](https://www.youtube.com/watch?v=-QssAt7Kx64) — 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-collimation-ab-50m-true-diff-0-250
- Whole chapter: https://www.bctengineer.com/md/semester_3/surveying_i/chapter-5

## Question

During permanent adjustment of collimation error of a level machine, the instrument was set up at F, mid of two staff stations A and B, 50 m apart, giving true elevation difference to be 0.250 m. During the 2nd setup 5 m from A and 45 m from B gives staff readings to be 1.500 and 1.752 m. Find the magnitude and direction of collimation error. Also calculate the staff readings at A and B for the 2nd setup corresponding to the horizontal line of sight. Also calculate the collimation precision ratio.

## Short answer

True difference: B lower by 0.250 m. Apparent: 1.752-1.500=0.252 m. Error=0.002 m over 40 m sight difference, error 0.00005 m/m (1/20000, 5 mm/100 m), line inclined upward. Correct readings: A=1.49975 m, B=1.74975 m.

## Answer

**Data:** A and B are 50 m apart. Mid-point setup F gives the difference of staff readings $B - A = 0.250$ m (B reads more, so B is lower than A by 0.250 m). Setup 2: 5 m from A and 45 m from B; readings A = 1.500 m, B = 1.752 m.

#### Step 1: Apparent difference and error

$$
\text{Apparent difference} = 1.752 - 1.500 = 0.252\ \text{m}
$$

$$
e = 0.252 - 0.250 = 0.002\ \text{m for the difference of sight lengths } (45 - 5) = 40\ \text{m}
$$

#### Step 2: Magnitude and direction of the collimation error

$$
\text{Error per metre} = \frac{0.002}{40} = 0.00005\ \text{m/m} = \frac{1}{20\,000}
$$

(that is 5 mm per 100 m, less than the usual limit of 10 mm per 100 m, so the instrument is within tolerance). The far staff B reads too high, so the line of collimation is **inclined upward** by about 10″ ($0.00005 \times 206265''$).

#### Step 3: Correct readings with a horizontal line of sight (setup 2)

Error at A (5 m): $0.00005 \times 5 = 0.00025$ m. Error at B (45 m): $0.00005 \times 45 = 0.00225$ m.

$$
A = 1.500 - 0.00025 = 1.49975\ \text{m}, \qquad B = 1.752 - 0.00225 = 1.74975\ \text{m}
$$

Check: $1.74975 - 1.49975 = 0.250$ m.

#### Step 4: Collimation precision ratio

$$
\text{Precision ratio} = \frac{e}{\text{difference of distances}} = \frac{0.002}{40} = \frac{1}{20\,000}
$$

**Answer: error = 0.00005 m per metre (1 in 20,000), line of collimation inclined upward; correct readings A = 1.49975 m (≈ 1.500), B = 1.74975 m (≈ 1.750).**

## Check yourself

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

1. Setup 2 is 5 m from A and 45 m from B. The distance difference used is:
   A. 40 m
   B. 50 m
   C. 45 m
   D. 5 m
   Answer: A. 45 - 5 = 40 m. (Key point: Sight difference)

2. Apparent difference is 1.752 - 1.500. Error is:
   A. 0.020 m
   B. 0.002 m
   C. 0.252 m
   D. 0.250 m
   Answer: B. 0.252 - 0.250 = 0.002 m over 40 m. (Key point: Error)

3. The precision ratio is:
   A. 1 in 40
   B. 1 in 125
   C. 1 in 20,000
   D. 1 in 2,000
   Answer: C. 0.002/40 = 0.00005 = 1/20000. (Key point: Precision)

4. The corrected readings are:
   A. A = 1.5025, B = 1.7745
   B. A = 1.498, B = 1.750
   C. A = 1.500, B = 1.752
   D. A = 1.49975, B = 1.74975
   Answer: D. Subtract 0.00005 x 5 = 0.00025 from A and 0.00005 x 45 = 0.00225 from B. (Key point: Corrected readings)

## Related in Surveying I

- [Collimation error with level mid-way and near B](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Atwo-peg-60m-staff-1-855-1-625) — Surveying I, chapter 5
- [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
- [Two peg: closing error and precision of level](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Atwo-peg-cd-50m-e-mid-f-5m) — 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
- [Two peg test and permanent adjustment](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Atwo-peg-test-procedure) — Surveying I, chapter 5, asked 4×
- [Collimation: level 5 m from B on AB produced](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Atwo-peg-50m-staff-1-875-1-790-b-5m) — Surveying I, chapter 5

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