Canals A and B join to form canal C, all having semi-circular cross-sections of radii which are in the ratio 3:4:5, respectively. Assume smooth merger of A and B, and ignore the possibility of flooding. If the speed s of water is the same and uniform in both A and B then the speed of water flowing in C is

Correct option is A

Given:

• semi-circular cross-sectionsCanals A, B, and C have .
• Radii of A, B, and C are in the ratio $3:4:5$.
• Speed of water in canals A and B is $s$.

We need to find the speed of water in canal C.

Concept:

The flow rate ($Q$) of water is conserved and is given by:

Flow rate = Cross-sectional area × Speed of water

For a semi-circular cross-section:
Area = (1/2) × π × r²

Step 1: Calculate the flow rates for canals A and B

1. Canal A:

• Radius = $3k$
• Flow rate = (1/2) × π × (3k)² × $s$
• Flow rate = (1/2) × π × 9k² × $s$

1. Canal B:

• Radius = $4k$
• Flow rate = (1/2) × π × (4k)² × $s$
• Flow rate = (1/2) × π × 16k² × $s$

Step 2: Total flow rate in canal C

The total flow rate in canal C is the sum of the flow rates in canals A and B:

Total flow rate = Flow rate of A + Flow rate of B
= (1/2) × π × 9k² × $s$ + (1/2) × π × 16k² × $s$

Factor out common terms:
Total flow rate = (1/2) × π × k² × $s$ × (9 + 16)
Total flow rate = (1/2) × π × k² × $s$ × 25

Step 3: Flow rate in canal C

For canal C:

• Radius = $5k$
• Flow rate = (1/2) × π × (5k)² × v_CvCv_C
  ​, where v_C vCv_C
  ​ is the speed of water in C.

Simplify:
Flow rate in C = (1/2) × π × 25k² × v_CvCv_C

Step 4: Equate total flow rate and flow rate in canal C

Equating both expressions for the flow rate:

(1/2) × π × k² × sss × 25 = (1/2) × π × 25k² × vCv_Cv_C

Cancel out common terms:
k2k²k₂, πππ, and 252525:

s=vCs = v_Cs=v_C

Final Answer:

The speed of water in canal C is equal to $s$.

Correct Option: (a) sss