Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2016 / S4 / 2

Neglect upstream speed and hydraulic losses. If $H$ is the upstream free-surface height above the crest, the specific energy at the crest is $H=y+v^2/(2g)$ and flow per unit width is $Q=yv$. At the critical broad-crested control section, extremizing $Q^2=2g{y}^2(H-y)$ gives $y=2H/3$ and $v^2=gy$. ThereforeFinite approach speed, friction and contraction introduce measured discharge coefficients; the ideal result assumes a long enough crest for the near-hydrostatic critical-flow approximation.

A Pelton wheel splits and turns a free jet through nearly 180 degrees in moving buckets, transferring momentum to the runner. For ideal bucket speed $V$, the absolute exit speed is $2V-u$ and force is $2\rho Q(u-V)$. Power $2\rho QV(u-V)$ is maximized at $V=u/2$, giving $P_{\max }=\rho Q{u}^2/2$. If the same inflow must now pass a slit of height $d$, continuity gives $u=Q/d$. Using the preceding flow rate with $H=y_{\min }$,The head must actually be sufficient to supply this jet speed: $u^2/(2g)$ cannot exceed the available hydraulic drop. An arbitrarily small slit cannot produce infinite power from a fixed-height reservoir; it would reduce discharge or raise water until another outlet, including overtopping, operates. The printed scaling presupposes compatible head and fixed-flow conditions.

Hydropower can convert a renewable water flux efficiently, has low operating fuel requirements, and reservoir schemes can provide rapid dispatch, inertia and storage. Long plant lifetimes can make capital expenditure worthwhile, while pumped storage shifts energy rather than producing a new primary resource.

Limitations include finite favorable sites, seasonal drought, sedimentation, habitat fragmentation, blocked fish migration, displaced populations and major civil-engineering/failure risks. Reservoir evaporation and methane emissions can matter, especially in warm flooded landscapes, so life-cycle emissions are not uniformly zero. Run-of-river plants avoid some reservoir impacts but offer less controllable output. A project assessment must compare useful annual energy and firm capacity, ecological and social costs, and the counterfactual electricity system.