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When you spray water from a garden hose you feel a backward force as you hold the nozzle. Similarly, anything you spray the water at feels a force in the direction away from the spray. What is the origin of this force, and what determines its magnitude?

Solution

Imagine a stream of water as a cylinder of uniform cross-sectional area A and density ρ. We consider an elemental unit of this that is Δx long. It travels, as does the rest of the stream, horizontally at velocity v (We will ignore the downward force of gravity here).

Consider the element of length Δx and area A and density ρ. Its mass m must therefore be

\begin{large}\[ m = \rho A \Delta x \] \end{large}

Since it travels with velocity v, its momentum is thus

\begin{large} \[ \vec{p} = \rho A \Delta x \vec{v} \] \end{large}

Let's assume that this element of the stream strikes an object (a wall, say), and breaks up, dissipating in all directions. the stream element loses all of its momentum in the process. The change in momentum of the stream element is thus

\begin{large} \[ \Delta \vec{p} = \rho A \Delta x \vec{v} \] \end{large}

If this happens in a time Δt, then the change in momentum with time (which is just the Average Force) is

\begin{large} \[ \vec{F_{avg}} = \frac{\Delta \vec{p}}{\Delta t} = \rho A \vec{v} \frac{\Delta x}{\Delta t} \] \end{large}

But Δx/Δt = v , so the magnitude of the Average Force is thus

\begin{large} \[ F_{avg} = \rho A v^{2} \]\end{large}