4.3 Time & Work, Pipes & Cisterns, Time, Speed & Distance

Time & Work, Pipes & Cisterns, Time, Speed & Distance

Efficiency analysis, project completion timelines, pipeline filling dynamics, and haulage transport logistics form the operational core of mining engineering. In CIL MT Paper-I, questions testing Time & Work and Time, Speed & Distance (TSD) are heavily emphasized. Mastering integer LCM models and relative speed mechanics allows candidates to bypass fraction-heavy systems of equations.

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1. Time & Work: The LCM Efficiency Model

Instead of assigning total work as $1$ and handling reciprocal fractions (e.g., $1/12 + 1/15$), assume total work to be the Least Common Multiple (LCM) of the individual completion times. This converts daily work rates into intuitive integer efficiencies.

Step-by-Step LCM Method

1. Define Total Work ($W$): $\text{LCM}(T_A, T_B, T_C)$.
2. Calculate Daily Efficiencies: $E_A = W / T_A$, $E_B = W / T_B$, $E_C = W / T_C$.
3. Calculate Combined Rate: $E_{\text{combined}} = E_A + E_B + E_C$.
4. Evaluate Completion Time: $T = \frac{W}{E_{\text{combined}}}$.

Alternate-Day Working Schedules

When workers operate in rotation (e.g., Worker A on Day 1, Worker B on Day 2):

1. Determine work accomplished during one complete cycle: $W_{\text{cycle}} = E_A + E_B$.
2. Determine number of complete cycles: $k = \lfloor W / W_{\text{cycle}} \rfloor$.
3. Calculate remaining work: $W_{\text{rem}} = W - (k \times W_{\text{cycle}})$.
4. Allocate $W_{\text{rem}}$ sequentially to the active worker in the next rotation.

Work and Wages Principle

Wages are divided strictly in proportion to the total quantum of work executed by each individual (Efficiency $\times$ Days Worked):

$\text{Wage Share Ratio } (W_A : W_B) = (E_A \times D_A) : (E_B \times D_B)$

Generalized Work Equivalence Formula

For problems with varying worker headcounts, daily hours, work efficiencies, and project scales, use the universal chain rule:

$\frac{M_1 \times D_1 \times H_1 \times E_1}{W_1} = \frac{M_2 \times D_2 \times H_2 \times E_2}{W_2}$

Where:

• $M = \text{Number of workers}$
• $D = \text{Number of days}$
• $H = \text{Working hours per day}$
• $E = \text{Relative worker efficiency}$
• $W = \text{Quantified work units (volume, length, items)}$

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2. Pipes & Cisterns Mechanics

Pipes and cisterns problems mirror Time & Work, with one critical extension: outlet pipes and leaks exert negative efficiency.

• Inlet Pipe ($A$): Fills tank in $T_A$ hours $\implies E_A = +\frac{W}{T_A}$
• Outlet Pipe / Leak ($B$): Empties tank in $T_B$ hours $\implies E_B = -\frac{W}{T_B}$
• Net Efficiency ($E_{\text{net}}$): $E_{\text{net}} = E_A - E_B$

$\text{Net Time to Fill } T = \frac{W}{E_A - E_B} = \frac{T_A \times T_B}{T_B - T_A} \quad (\text{when } T_B > T_A)$

3. Time, Speed & Distance (TSD) Fundamentals

Core Formulas & Unit Conversions

$\text{Distance } (D) = \text{Speed } (S) \times \text{Time } (T)$

• km/h to m/s: Multiply by $\frac{5}{18}$ (e.g., $72\text{ km/h} = 72 \times \frac{5}{18} = 20\text{ m/s}$).
• m/s to km/h: Multiply by $\frac{18}{5}$ (e.g., $15\text{ m/s} = 15 \times \frac{18}{5} = 54\text{ km/h}$).

Average Speed

$\text{Average Speed} = \frac{\text{Total Distance Traveled}}{\text{Total Time Taken}}$

• Two Equal Distance Segments: When an object travels equal distances at speeds $S_1$ and $S_2$, the average speed is the harmonic mean: $S_{\text{avg}} = \frac{2 S_1 S_2}{S_1 + S_2}$
• Three Equal Distance Segments: $S_{\text{avg}} = \frac{3 S_1 S_2 S_3}{S_1 S_2 + S_2 S_3 + S_3 S_1}$

Late and Early Arrival Distance Shortcut

If traveling at speed $S_1$ leads to arriving $t_1$ minutes late, and traveling at speed $S_2$ leads to arriving $t_2$ minutes early (net time difference $\Delta t = t_1 + t_2$ hours):

$D = \frac{S_1 \times S_2}{|S_1 - S_2|} \times \Delta t$

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4. Trains, Relative Velocity, and Boats & Streams

Train Crossing Formulas

Boats and Streams Mechanics

Let speed of boat in still water $= u\text{ km/h}$ and speed of water current $= v\text{ km/h}$:

• Downstream Speed ($D$): Motion with the current $\implies D = u + v$
• Upstream Speed ($U$): Motion against the current $\implies U = u - v$

From these two equations, we obtain:

$\text{Speed of Boat in Still Water } (u) = \frac{D + U}{2}$ $\text{Speed of Current } (v) = \frac{D - U}{2}$

• Round-Trip Average Speed: $S_{\text{avg, round trip}} = \frac{2 \times D \times U}{D + U} = \frac{2(u + v)(u - v)}{2u} = \frac{u^2 - v^2}{u}$