Instrument near P: staff at P = 1.800 m, staff at Q = 2.600 m. Instrument near Q: staff at P = 1.500 m, staff at Q = 2.400 m. RL of P = 100 m. Find RL of Q.
Correct Answer: A. 99.15 m
📚 Detailed Explanation: Reciprocal Levelling — RL of Q = 99.15 m
Why A (99.15 m) is correct: Reciprocal levelling is used when the two points are far apart and the instrument cannot be set at the midpoint. By taking readings from both ends, errors due to curvature, refraction, and collimation are automatically cancelled. The true height difference is the average of the two apparent differences.
Reciprocal Levelling Formula
True height difference:
h = [(Staff at P from setup near P) – (Staff at Q from setup near P)
+ (Staff at P from setup near Q) – (Staff at Q from setup near Q)] / 2
h = [(Staff at P from setup near P) – (Staff at Q from setup near P)
+ (Staff at P from setup near Q) – (Staff at Q from setup near Q)] / 2
Given:
Setup near P: staff at P = 1.800 m, staff at Q = 2.600 m
Setup near Q: staff at P = 1.500 m, staff at Q = 2.400 m
h = [(1.800 – 2.600) + (1.500 – 2.400)] / 2
h = [(-0.800) + (-0.900)] / 2
h = -1.700 / 2
h = -0.850 m (negative → Q is LOWER than P)
RL of Q = RL of P + h = 100.000 + (-0.850) = 99.150 m
Why Both Setups Give Different Apparent Differences
| Error Type | Setup Near P | Setup Near Q | Cancels in Average? |
|---|---|---|---|
| Collimation error | Affects long sight to Q more | Affects long sight to P more | ✓ YES |
| Earth curvature | Larger error on Q (farther) | Larger error on P (farther) | ✓ YES |
| Atmospheric refraction | Larger on distant Q | Larger on distant P | ✓ YES |
- True h = average of two apparent differences: (−0.800 + −0.900)/2 = −0.850 m
- RL of Q = 100 + (−0.850) = 99.15 m
- Reciprocal levelling eliminates collimation error, curvature error, and refraction error simultaneously.