Jet Power from a Nozzle – Optimal Pipe Friction Loss For a nozzle fed by a long pipe from a supply head, the power delivered by the exit jet is maximum when the head lost due to pipe friction equals one-third of the total supply head.

Introduction:
Maximizing jet power at a nozzle connected to a reservoir through a pipe involves trading off discharge (which increases with frictional drop) against delivery head (which decreases as losses rise). The classical result identifies the best split for maximum power output.

Given Data / Assumptions:

• Total supply head H available at pipe inlet.
• Head loss due to friction h_f along the pipe.
• Jet exit head H_d = H − h_f; jet power proportional to Q * H_d.

Concept / Approach:

Under typical assumptions, Q increases with h_f (since more head is spent overcoming friction, velocity and Q adjust). Optimizing P = rho * g * Q * (H − h_f) with respect to h_f yields the one-third rule for maximum power transfer to the jet.

Step-by-Step Solution:

Verification / Alternative check:

Substitute h_f = H/3 and check that small deviations reduce P on both sides (second-derivative negative), confirming a maximum.

Why Other Options Are Wrong:

One-half, two-third, one-fourth, three-fourths split do not satisfy the optimum condition and yield lower power.

Common Pitfalls:

Confusing efficiency with maximum power; ignoring nozzle losses or minor losses, which shift numerical details but not the qualitative one-third insight.

Final Answer:

one-third