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Not really sure if what I'm doing is right. Presumably it depends on the angle of your arm in the water. If we let the speed of the water at your arm be V, we get, by Bernoulli, taking the height up your arm to be h, the pressure to be p_atm before and p afterwards:

p_atm/ρ + U²/2 = p/ρ + V²/2.

And we know p = p_atm + ρgh.

Which gives:

U²/2 = gh + V²/2.

I'm trying to use mass conservation now, but then don't we need the cross-sectional area of the stream, and the area that is occupied by your arm?

Am I going about the question in completely the wrong way?

Help please :) !