---
title: "Water jet on vane moving at 16 m/s"
subject: "Applied Mechanics (Dynamics)"
chapter: 3
key: "semester_3/applied_mechanics_dynamics:nozzle-fixed-vane-moving-4000mm2"
source: https://www.bctengineer.com/q/semester_3/applied_mechanics_dynamics/nozzle-fixed-vane-moving-4000mm2
site: BCT Engineering
---

# Water jet on vane moving at 16 m/s

Applied Mechanics (Dynamics) · Chapter 3: System of particles

- Asked in: 2076 Asoj, 5+3 marks
- Video: [FLUIDME: Force Upon a Moving Vane](https://www.youtube.com/watch?v=5uHVOtOF2NA) — MES DLSU
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/applied_mechanics_dynamics/nozzle-fixed-vane-moving-4000mm2
- Whole chapter: https://www.bctengineer.com/md/semester_3/applied_mechanics_dynamics/chapter-3

## Question

A nozzle discharges a stream of water of cross sectional area $A = 4000$ mm$^2$ with a speed $v = 48$ m/sec, and the stream is deflected by a fixed vane which is moving in the same direction of water flow with constant speed of 16 m/sec as shown in figure. The mass density of water $\rho = 1000$ kg/m$^3$. Determine the resultant force exerted on the stream by the fixed vane and maximum power developed. [Figure: stream at 48 m/s toward a vane B moving at 16 m/s in the same direction; the stream leaves the vane at C at $60^\circ$ to the horizontal.]

## Short answer

Relative velocity 32 m/s, mass flow rate 128 kg/s. Momentum change gives resultant force on the stream 4096 N (horizontal component -2048 N, vertical 3547 N). Power developed is 32.8 kW.

## Answer

**Assumptions:** the vane moves at $V=16$ m/s in the direction of the jet; the stream leaves at C at $60^\circ$ above the horizontal relative to the vane; friction is neglected, so the relative speed is unchanged along the vane.

#### Relative velocity and flow rate
$$
v_{rel}=v-V=48-16=32\ \text{m/s}
$$

The mass of water striking a single moving vane per second is

$$
\dot m=\rho A(v-V)=1000(4000\times10^{-6})(32)=128\ \text{kg/s}
$$

#### Force on the stream
Apply the momentum equation in the frame moving with the vane (constant velocity, so Newton's laws hold):

$$
\vec F=\dot m(\vec v_{C,rel}-\vec v_{B,rel})
$$

$$
\begin{aligned}
F_x&=128\left(32\cos60^\circ-32\right)=-2048\ \text{N}\\
F_y&=128\left(32\sin60^\circ\right)=3547\ \text{N}\\
F&=\sqrt{2048^{2}+3547^{2}}=4096\ \text{N}
\end{aligned}
$$

This is the force the vane exerts on the stream. The stream exerts an equal and opposite force on the vane: $2048$ N forward (along the jet) and $3547$ N downward.

#### Power developed
Only the force component in the direction of the vane's motion does work:

$$
P=F_xV=2048(16)=32\,768\ \text{W}\approx32.8\ \text{kW}
$$

For a single moving vane, $P\propto(v-V)^2V$, and $dP/dV=0$ gives $V=v/3=16$ m/s. The given vane speed is exactly this value, so this is the maximum power.

**Answer: resultant force on the stream $=4096$ N (components $-2048$ N and $3547$ N); maximum power $\approx32.8$ kW.**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What speed is used for water relative to the vane?
   A. 48 m/s
   B. 64 m/s
   C. 16 m/s
   D. 32 m/s
   Answer: D. v_rel = 48 - 16 = 32 m/s. (Key point: Relative velocity)

2. Why is m dot = rho A (v - V) = 128 kg/s rather than rho A v?
   A. Only water reaching the moving vane is counted, and the vane moves away at V
   B. The vane is heavier
   C. Water slows down in the nozzle
   D. Density changes at the vane
   Answer: A. A single moving vane receives only the water that catches up with it, at relative speed v - V. (Key point: Flow rate on single vane)

3. Which force component gives the power, and what is it?
   A. F_y = 3547 N, giving 56.8 kW
   B. Resultant 4096 N, giving 65.5 kW
   C. F_x = 2048 N times V = 16 m/s, about 32.8 kW
   D. F_x times 48 m/s
   Answer: C. Only the component along the vane's motion does work: P = 2048(16) = 32.8 kW. (Key point: Power)

4. Why is this the maximum power?
   A. The vane speed equals the jet speed
   B. The vane is at rest
   C. The angle is 90 degrees
   D. V = v/3 is the vane speed that maximises power of a single moving vane
   Answer: D. Maximising (v - V)^2 V gives V = v/3 = 16 m/s. (Key point: Maximum power)

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