Reciprocal levelling across a pond: From setup near A — staff at A = 2.5 m, staff at B = 2.0 m. From setup near B — staff at A = 1.2 m, staff at B = 1.7 m. True height difference A and B (m):
Correct Answer: B. 0.5 m
📚 Detailed Explanation: Reciprocal Levelling Across Pond — Difference = 0.5 m
Why B (0.5 m) is correct: The true height difference between A and B is the average of the two apparent differences observed from opposite sides of the pond. Both apparent differences are 0.5 m; their average is also 0.5 m.
Setup near A:
Staff at A = 2.5 m (BS), Staff at B = 2.0 m (FS)
Apparent diff from A = BS(A) – FS(B) = 2.5 – 2.0 = 0.5 m (B appears higher)
Staff at A = 2.5 m (BS), Staff at B = 2.0 m (FS)
Apparent diff from A = BS(A) – FS(B) = 2.5 – 2.0 = 0.5 m (B appears higher)
Setup near B:
Staff at A = 1.2 m (FS), Staff at B = 1.7 m (BS)
Apparent diff from B = BS(B) – FS(A) = 1.7 – 1.2 = 0.5 m (B appears higher)
True height difference:
h = (0.5 + 0.5) / 2 = 1.0 / 2 = 0.5 m
B is 0.5 m higher than A.
Why Both Apparent Differences Are the Same Here
| Consideration | Result |
|---|---|
| Both apparent diffs equal 0.5 m | No systematic error in this case (collimation and refraction errors are equal in both setups and cancel in the average) |
| True diff = average | 0.5 m (B is higher than A) |
| If apparent diffs were different | Average would still give the true diff; any asymmetric error would cancel |
- Apparent diff from A: 2.5 − 2.0 = 0.5 m.
- Apparent diff from B: 1.7 − 1.2 = 0.5 m.
- True diff = (0.5 + 0.5)/2 = 0.5 m.