Work-Rate, Pipes & Cisterns, Relative Speed & Motion
“LCM efficiency method, negative work of drainage pipes, relative velocity, trains, and river boats.”
1.The Efficiency Method and Kinematic Frames
§ 1.1.Instead of manipulating fractions like 1/12 + 1/15 - 1/20 under exam time constraints, the LCM Unit Efficiency Framework eliminates fractions completely.
Relative Motion Principle: To analyze two moving bodies, fix one body at rest by subtracting its velocity vector from the other body. Distance covered is the initial separation between the bodies plus their physical lengths.
Upstream vs Downstream Mechanics:
- Downstream: River current assists motion: v_d = u + v.
- Upstream: River current opposes motion: v_u = u - v.
- Boat in still water: u = (v_d + v_u) / 2.
- Stream velocity: v = (v_d - v_u) / 2.
If downstream speed is 16 km/h and upstream speed is 10 km/h, still water speed is simply (16 + 10)/2 = 13 km/h, and water current speed is (16 - 10)/2 = 3 km/h.
2. Worked Problem: Alternating Work with Pipe Leak
When working in alternate hours, do not divide total work by cycle work directly near the end, because the tank might fill during an active filling hour before the leak hour occurs.
Problem [Original Practice Problem · Modeled on GATE Pattern]:
Pipe A can fill a tank in 12 hours. Pipe B empties the full tank in 18 hours.
The pipes are opened on alternate hours starting with Pipe A. How many hours will it take for the tank to be completely filled?
Step 1: Set Total Capacity = LCM(12, 18) = 36 units:
Efficiency(A) = +36 / 12 = +3 units/hour.
Efficiency(B) = -36 / 18 = -2 units/hour.
Step 2: Analyze a 2-hour cycle (Hour 1: A, Hour 2: B):
Work done in 2 hours = 3 - 2 = +1 unit.
Step 3: Guard against overflow near the end:
Since A can contribute +3 units in a single hour, stop the cycle analysis before reaching (36 - 3) = 33 units.
To reach 33 units: 33 cycles of 2 hours each = 33 * 2 = 66 hours.
After 66 hours, work completed = 33 units.
Step 4: Analyze remaining work on Hour 67 (Pipe A's turn):
Remaining work = 36 - 33 = 3 units.
Pipe A does exactly +3 units in 1 hour.
Therefore, on Hour 67, the tank becomes completely full and the process terminates.
Total time taken = 66 + 1 = 67 hours.
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