GSO002 Problem 5

This problem tests understanding of the relationship between momentum change and impulse for a fluid.

1. Mass Hitting the Plate in a Small Time Interval

Consider a small time interval $\Delta t \ [\text{s}]$ . The water ejected from the nozzle during $\Delta t$ travels a distance $v \Delta t$ . The volume $\Delta V$ of water that collides with the plate is:

\Delta V = S \cdot (v \Delta t)

The mass $\Delta m$ of this water is:

\Delta m = \rho \Delta V = \rho S v \Delta t

2. Impulse from Momentum Change

The $x$ -direction velocity of the water changes from $v$ before the collision to $0$ after.

By Newton's third law, the impulse $\Delta I$ received by the plate from mass $\Delta m$ is:

\Delta I = \Delta m \cdot v = \rho S v^2 \Delta t

3. Average Force on the Plate

The average force $F$ on the plate is:

F = \frac{\Delta I}{\Delta t} = \rho S v^2

4. Numerical Calculation

Substituting the given values:

\begin{aligned} F &= (1.0 \times 10^3) \times (1.25 \times 10^{-6}) \times (640)^2 \\ &= 1.25 \times 10^{-3} \times 409600 \\ &= 1.25 \times 409.6 \\ &= 512 \ \text{N} \end{aligned}

This force is applied continuously for $T = 8.0 \ \text{s}$ , so the total impulse $I$ is:

\begin{aligned} I &= F \times T \\ &= 512 \times 8.0 \\ &= 4096 \ \text{N}\cdot\text{s} \end{aligned}

Answer: 4096

GSO002 Problem 5

Impact Force of an Ultra-High-Speed Water Jet Cutter

Problem Statement

Constraints

Input Format

Solution

1. Mass Hitting the Plate in a Small Time Interval

2. Impulse from Momentum Change

3. Average Force on the Plate

4. Numerical Calculation