A distance of 2000 meters was measured by a 30 meter chain. Later on, it was detected that the chain was 0.1 meter too long. Another 500 meter (i.e., total 2500 m) was measured and it was detected that the chain was 0.15 meter too long. If the length of the chain in the initial stage, was quite correct, determine the exact length that was measured.

Key Information

• Nominal Length of Chain (L_nominal) = 30 m
• Initial Actual Length (L_start) = 30.00 m (Error = 0 m)
• Actual Length after 2000 m (L_mid) = 30 m + 0.10 m = 30.10 m (Error = +0.1 m)
• Actual Length after 2500 m (L_end) = 30 m + 0.15 m = 30.15 m (Error = +0.15 m)
• Measured Distance Segment 1 = 2000 m
• Measured Distance Segment 2 = 2500 m – 2000 m = 500 m
• Goal: Find the Total True Distance measured.

Step 1: Calculate True Distance for First Segment (0 m to 2000 m)

Step 2: Calculate True Distance for Second Segment (2000 m to 2500 m)

Step 3: Calculate Total True Distance

Final Result

Core Concepts:

1. Linear Measurement & Systematic Error: Measuring distance using tools like chains can introduce systematic errors if the tool’s actual length deviates from its nominal length.
2. Variable Error / Calibration Drift: The error is not constant; the chain stretches or changes length during the measurement process due to use, temperature, or other factors.
3. Segmented Correction Strategy: When the error changes and intermediate checks are available, the total measurement must be divided into segments. The correction is calculated separately for each segment.
4. Assumption of Linear Error Change: Within each segment (between checks), the most common approach is to assume the error changes linearly. This allows using the average of the start and end errors (or lengths) for that segment.
5. Average Length Correction: The correction for each segment uses the formula: True Distance = (Average Actual Length / Nominal Length) × Measured Segment Length.

Real-World Applications:

• Land Surveying: Crucial for maintaining accuracy in boundary surveys, topographic mapping, and construction stakeout, especially over long distances or durations where instrument calibration might drift.
• Civil Engineering Projects: Ensuring precision in setting out roads, railways, pipelines, and structures, where cumulative errors can be significant.
• Analysis of Historical Surveys: Reconstructing accurate measurements from old field notes that include records of instrument checks.
• Quality Assurance in Measurement: Implementing procedures that include periodic checks and corrections for instruments known to experience drift.
• Geodetic Surveys: In high-precision surveys, accounting for all potential error sources, including changes in instrument calibration during the survey, is standard practice.

Why It Works:
Applying a single correction based on either the intermediate (0.1m too long) or final (0.15m too long) error would misrepresent the actual conditions during measurement. The chain was correct initially, then progressively elongated. Segmenting the measurement (0-2000m and 2000-2500m) allows us to isolate periods where the error change was limited. By assuming a linear increase in error within each segment, we can estimate the average effective length of the chain during that period (30.05m for the first 2000m, 30.125m for the next 500m). Applying the correction factor (Average Actual Length / Nominal Length) to the measured length *for each segment* accounts for the average state of the chain during that specific part of the measurement. Summing these corrected segment lengths provides the best estimate of the total true distance covered, reflecting the gradual change in the chain’s length.