Problem Statement
A stream of water of cross-section A flows at speed v and strikes a flat vertical wall perpendicularly. The water does not bounce back. Find the force on the wall.
Given Information
- All numerical data are stated in the problem above; symbols are defined as they appear.
Physical Concepts & Formulas
These problems use the continuity equation and Bernoulli’s theorem for an ideal incompressible fluid, plus viscosity where stated. All momentum destroyed.
- $S_1 v_1 = S_2 v_2$ — continuity equation
- $p + \rho g h + \tfrac{1}{2}\rho v^2 = \text{const}$ — Bernoulli’s equation
- $F = 6\pi\eta r v$ — Stokes’ law
Step-by-Step Solution
Step 1 — Identify the governing principle: We begin by recognising which physical law controls the situation and why it is the correct starting point for this problem.
$$Volume flow rate = Av$$
Step 2 — Set up the relevant equations: Next we write down the equations that follow from that principle, introducing the symbols we will carry through the algebra.
$$Mass flow rate = \rho Av$$
Step 3 — Apply the given conditions: We now substitute the specific conditions and constraints given in the problem so the equations describe this particular situation.
$$Momentum delivered per second = \rho Av \cdot v = \rho Av^{2}$$
Step 4 — Solve for the required quantity: With the equations specialised, we isolate and solve for the unknown the problem asks us to find.
$$F = \rho Av^{2}$$
Worked Calculation
$$F = \rho Av^{2}$$
Answer
$$\boxed{F = \rho Av^{2}}$$
This is the quantity the problem asked for, expressed in terms of the given data: $F = \rho Av^{2}$.
Physical Interpretation
This is the dynamic pressure force — the same principle behind jet propulsion, hose nozzles, and wind loading on buildings. The magnitude of the answer is consistent with everyday physical experience for this class of problem in Irodov’s Part 1 — the result shows how the answer scales with the given quantities. If we doubled the dominant input, the boxed formula tells us exactly how the output would respond, and that scaling is the key physical insight this problem trains. Comparing the answer with the appropriate limiting cases (very small or very large values of the dominant parameter) recovers the familiar Newtonian or intuitive expectation, which is a useful sanity check.
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