Why do the cesium atoms in an atomic clock have to be tossed into the air like a fountain? Even the NIST website for the standard F-1 clock and the new F-2 do not explain, exactly, why it helps (or is necessary) to make the cesium atoms appear, at least temporarily, weightless....
Wouldn't their absorption and emissions be redshifted and blueshifted by the motion?
 A: 
Wouldn't their absorption and emissions be redshifted and blueshifted by the motion?

The motion is perpendicular to the microwave cavity, so large Doppler shifts are not observed relative to that direction due to the tossing up of atoms. Although, the velocity always has some uncertainty both perpendicular and parallel to the cavity, and the angle of the trajectory relative to the microwave cavity never precise, this will broaden the linewidth.
The main reason why atoms in an atomic fountain clock or fountain configuration are projected up is to increase the free-evolution time and simultaneously decrease the size of the device and also reduce unwanted noise. The reasoning for these are as follows:

*

*During the free evolution, a fictitious phase $\phi=-\delta t$ is accumulating, where $\delta=\omega-\omega_e$ is the detuning of the microwave $\omega$ from the excited state transition $\omega_e$. This phase is fictitious because there is no interaction between the microwave field and atoms. However, when the atoms fall back down into the cavity they do interact and the action depends on $\phi$. Larger observation time $t$ leads to a larger phase shift and tossing the atoms up in the air increases $t$.

*If you were to try and get the same observation time through an apparatus that was placed, for example, horizontally, it would have to be very long and deep. Long to accommodate the high velocity of the atoms and deep to also compensate for the acceleration of the atoms under gravity. So having the atoms tossed up, reduced the overall size of the device.

*Having a single microwave cavity ensures coherence between the pulses. Having separate cavities can result in unwanted phase noise being imprinted in the readout signal.

A: According to the description on the nist.gov website of the NIST-F1 Cesium Fountain Atomic Clock the atomic fountain setup was chosen to obtain a longer observation window. According to the article the atoms have a round trip time of a second.
Quote:

Traditional cesium clocks measure room-temperature atoms moving at several hundred meters per second. Since the atoms are moving so fast, the observation time is limited to a few milliseconds.

Summerizing:
As I understand that description: the atomic fountain setup is used for two reasons:
-the velocities of the observed cesium atoms are much slower than is the case with what is in the article referred to as traditional cesium clock setup.
-the observation time is much longer
