Will a cut shoot (as is used in a potometer for measuring transpiration) take up water more quickly than a rooted plant?

The xylem is a tortuous pathway, with water being transferred across "pit membranes" at the base of "pits", the "holes" which can be seen in micrographs of vessels and tracheids of conducting tissue; the water flow between tracheids is necessairly tortuous as the narrow cells overlap and so water must transfer across a set of these membrane from cell to cell: vessels consist of maybe 5 or 10 cells bolted together to make a continuous "tube" of some 500 to 50,000 micrometers (5 cm!), before a transverse endplate with pits and pit membranes seals off each compartment. Remember gymnosperms - pines etc - only have overlapping tracheids, 100 - 500 micrometers in length, with such a tortuous pathway for water flow, and that is why they are outcompeted by angiosperms unless it is cold in summer and winter (boreal forests) or at high altitude.

So xylem is a high resistance pathway, and it starts in the roots. So in theory, if you cut off the roots, and recut the stem under water to exclude any air bubbles drawn into the xylem by the tension of the transpiration stream, you have reduced the resistance of the overall pathway from soil, via roots to shoots. Conclusion: the flux in a potometer should be higher in a shorter stem!

So I would agree with your conclusion, that water uptake should be increased due to the direct access to xylem vessels in a cut stem, rather than  resistance to this accross the apoplast in a rooted plant.

However, we also know that a weak solution (say 50 mM KCl as sap) flows faster than distilled /deionised water... because the pectin gels lining the pores of the pit membranes shrink or swell depending on the osmotic strength of the xylem "sap". So the previous conclusion relies on you using a weak ionic solution and recutting the stem under water (the stem, not you as well!) to remove any air bubbles.

Prof Howard Griffiths, University of Cambridge