Momentum Transport

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

System:

Element of water stream as a point particle acting on an article, also treated as a point particle.

Interactions:

We will ignore the vertical direction, so that the only interaction is the force due to the changes in momentum of the water and the item hit..

Model:

Momentum and External Force.

Approach:

Diagrammatic Representation

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).

Mathematical Representation

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

Since it travels with velocity v, its momentum is thus

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

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

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