A jet of water strikes a flat plate perpendicularly, exerting force F on the plate. The plate is replaced by a curved bucket that deflects the same jet exactly 180°. With identical jet velocity and flow rate, the force on the bucket is:
AThe same force F — mass flow rate and inlet velocity are unchanged
BHalf of F — only the component of velocity parallel to the jet axis contributes
CTwice F — the jet momentum completely reverses, doubling the total change in momentum
DFour times F — force depends on velocity squared, and reversing velocity is equivalent to doubling it
For a flat plate perpendicular to the jet, the flow exits sideways with zero x-momentum, so the force equals the incoming momentum flux: F = ṁV = ρAV². For a 180° deflecting bucket, the jet exits in the reverse direction with x-momentum −ṁV, giving a total momentum change of 2ṁV and force 2ρAV² — exactly twice as large. The general formula is F = ρAV²(1 − cosθ), which gives 2ρAV² at θ = 180°. Option D confuses the squaring from the momentum flux formula (ρAV²) with what changes when deflection angle changes.
Question 2 Multiple Choice
A Pelton wheel bucket moves at speed u = 10 m/s and intercepts a water jet with absolute velocity V = 30 m/s. What velocity governs the momentum exchange between the jet and the bucket?
AV = 30 m/s, the absolute jet velocity, because momentum is always measured in a fixed frame
BV − u = 20 m/s, the relative jet velocity, because the bucket only intercepts momentum not already matched by its own motion
CV + u = 40 m/s, the sum of both velocities, because the bucket moves toward the oncoming jet
Du = 10 m/s, the bucket velocity, because power equals force times velocity and only bucket speed matters for power
When a surface moves, the momentum exchange depends on the relative velocity between the jet and the surface. The bucket receives fluid at relative velocity (V − u) and must redirect it. Substituting into the force formula: F = ρA(V − u)²(1 − cosθ). When u = 0 (stationary), the full jet velocity applies. When u = V (bucket moving at jet speed), the relative velocity is zero and no force is exerted — the jet just touches the bucket without exchanging momentum. Only the relative motion drives momentum transfer, just as with any impact problem in classical mechanics.
Question 3 True / False
For a curved vane deflecting a jet by angle θ, the force on the vane increases monotonically as θ increases from 0° to 180°, reaching its maximum when the jet is fully reversed.
TTrue
FFalse
Answer: True
The force formula F = ρQV(1 − cosθ) confirms this directly. At θ = 0° (no deflection), cos 0° = 1 and F = 0 — the jet passes through unchanged. At θ = 90° (perpendicular deflection), cos 90° = 0 and F = ρQV. At θ = 180° (full reversal), cos 180° = −1 and F = 2ρQV — the maximum possible force. Since (1 − cosθ) is monotonically increasing on [0°, 180°], the force increases continuously with deflection angle.
Question 4 True / False
A flat plate perpendicular to a water jet exerts the same force as a curved bucket that deflects the same jet by 180°, because in both cases the jet's kinetic energy is fully removed.
TTrue
FFalse
Answer: False
The forces are not equal — the 180° bucket exerts twice the force of the flat plate. For the flat plate, the jet exits sideways with zero axial momentum, so the axial force equals ṁV. For the 180° bucket, the jet exits in the reverse direction with axial momentum −ṁV, so the total change in axial momentum is 2ṁV — double the force. Force depends on momentum change, not energy removal. (The flat plate also removes kinetic energy but does so by redirecting it sideways, not reversing it.)
Question 5 Short Answer
Why are Pelton wheel buckets designed to deflect the incoming water jet as close to 180° as possible, rather than using flat plates or smaller deflection angles?
Think about your answer, then reveal below.
Model answer: The force exerted on a surface by a jet equals the rate of change of momentum of the fluid. For a flat plate (perpendicular impact), the jet exits sideways with zero axial momentum, so the force equals ṁV. For a 180° deflection, the jet reverses direction and the total axial momentum change is 2ṁV — twice as large. Maximum force means maximum torque on the wheel and maximum power extracted from the water. Pelton buckets approach hemispherical shapes to reverse the jet as completely as possible, with a slight opening to prevent the exiting water from striking the next bucket.
In practice, buckets are designed with a deflection angle slightly less than 180° (typically about 165°–170°) to prevent interference between the exiting flow and the next bucket in line. The small deviation from 180° causes only a small reduction in force (since cos 165° ≈ −0.97, giving F ≈ 1.97ρAV² instead of 2ρAV²), so the design target remains as close to full reversal as mechanically feasible.