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Photo from Wikimedia Commons |
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:
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:
Approach:
Diagrammatic Representation
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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
\begin
[ m = \rho A \Delta x ] \end
Since it travels with velocity v, its momentum is thus
\begin
[ \vec
= \rhp A \Delta x \vec
] \end
xxxxxxxx
All that remains is to determine the mass of the rain. We can do this by noting that the density of water is 1000 kg/m3 and that the water has filled the car to 2.0 cm deep, indicating a collected mass of:
\begin
[ m^
= \rho^
V^
= (\mbox
^
)(\mbox
\times\mbox
\times\mbox
) = \mbox
]\end
where ρwater is the density of water and Vrain is the volume of the accumulated rain.
We can now solve to find:
\begin
[ v_
= \frac{m^
v^
_{x,i}}{m^
+ m^{\rm rain}} = \mbox
]\end