Levelling on an Up Gradient – Limiting Sight Length with Staff and Instrument Height In differential levelling, with instrument height 1.0 m above ground, staff length 4.0 m, and the ground rising at 1 in 10, what is the maximum permissible sight distance on the up slope so that the line of sight still intersects the staff scale (i.e., staff reading remains non-negative)?

Introduction:
On rising ground, a horizontal line of sight progressively approaches the ground. With a finite instrument height (HI) and a finite staff length, the sight distance must be limited so that the horizontal line of sight still strikes the staff (returns a non-negative reading). This question applies simple geometry to find that limiting distance.

Given Data / Assumptions:

• Instrument height HI = 1.0 m above ground at the instrument station.
• Staff length = 4.0 m (sufficiently long for typical readings).
• Up gradient = 1 in 10 (rise = 1 m per 10 m horizontally).

Concept / Approach:

For a horizontal line of sight over distance x on an up gradient 1 in 10, the ground at the staff point is higher than the instrument point by x/10. The height of the line of sight above the ground at the staff point equals HI − x/10. The reading becomes zero when HI − x/10 = 0; beyond that distance the line of sight would hit the ground below the staff bottom.

Step-by-Step Solution:

Verification / Alternative check:

Staff length (4 m) is not the limiting factor on an up-slope sight because readings decrease with distance; the binding constraint is h_sight ≥ 0.

Why Other Options Are Wrong:

20 m and 45 m exceed the geometric limit; 25 cm and 5 m are unnecessarily conservative given the HI and gradient.

Common Pitfalls:

Using staff length as the limiting criterion on an up gradient (it mainly limits down-slope sights where readings increase).

Final Answer:

10 m