You have probably observed that a steady stream of water flowing out
of a faucet gets smaller the farther it gets from the faucet.  Why
does this happen? Explain in less than 50-words.

Hello.

Here's a 50 word explanation:

The water from the faucet is falling. As it falls, it speeds up (at
least over short distances).  If we have a higher velocity at the
bottom of the stream, the cross-sectional size of the stream must be
smaller in order for the flow rate to remain the same.


sources:

"The water that emerges from the faucet is falling. What happens to
any object that falls under the influence of gravity? It travels
faster the further it falls (at least over short distances)...   One
can understand that if we have a higher velocity at the bottom of the
stream, then the cross-section... is going to have to be smaller in
order for the flow rate to remain the same. Thus the size of the
stream... gets smaller the further (and faster) the water falls. If
the stream falls far enough, the water reaches a terminal speed and
the size of the stream will stop decreasing in size or becoming
smaller as it falls."
http://www.uu.edu/programs/physics/sciguysjune2001.htm

Also see:

" For example, if you've ever noticed water flowing from a faucet set
open to give a relatively small flow of water, you might have noticed
that the flow narrows as the water falls away from the faucet opening.
 If we look at the flux of water, defined as the amount or mass of
water flowing across an area, A1, per unit of time, then we know that
the same mass of water per unit time must flow across A2, placed at a
lower position. However, since gravity causes the speed of the water
to increase as it falls, the cross-sectional area of the water, that
is to say, the size of the stream going through A1 must be larger than
the size of the stream going through A2 since the volume of water (and
hence the total mass since the water has constant density) per unit
time crossing A1 or A2 is given by (cross-sectional area) ×
(velocity). Hence, since velocity goes up, cross-sectional area must
go down. Note that water reaches its terminal velocity quite quickly
so that the stream reaches a constant cross-sectional area after just
a small distance of travel."
http://dept.physics.upenn.edu/courses/gladney/phys151/lectures/lecture_jan_22_2003.shtml

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search strategy:
faucet "stream narrows"
"Continuity Equation" faucet

I hope this helps.