That makes it more difficult :). Would you be satisfied if I used some plausible-sounding numbers and let you extrapolate from there? Otherwise there are a lot of constants to keep track of and the post editor isn’t great for formulas.
Let’s look at the stick-ball collision first. We’ll model this as a perfect elastic collision (it’s not, but it’s pretty close). The formula for the final velocity of the impacted object (which you can look up on Wikipedia) is:
v[sub]2[/sub] = (u[sub]2[/sub](m[sub]2[/sub] - m[sub]1[/sub]) + 2m[sub]1[/sub]u[sub]1[/sub]) / (m[sub]1[/sub] + m[sub]2[/sub])
u[sub]2[/sub] is the initial velocity of the ball, which is 0. I’ll also plug in 0.16 kg for m[sub]2[/sub].
v[sub]2[/sub] = 2m[sub]1[/sub]u[sub]1[/sub] / (m[sub]1[/sub] + 0.16 kg)
All right, so we have v[sub]2[/sub] in terms of the mass of the stick and stick velocity. We want to reduce this to just the stick mass.
The stick is accelerated by the arm, which can accelerate the arm+stick through a fixed distance. Let’s say that the arm can exert 150 N of force through 0.5 m distance and has 2 kg of effective swinging arm mass. F=ma, so:
150 N = m[sub]1[/sub]a
a = 150 / (m[sub]1[/sub] + 2)
We also have d = 0.5at[sup]2[/sup]:
0.5 m = 0.5at[sup]2[/sup]
a = 1/t[sup]2[/sup]
Setting equal:
1/t[sup]2[/sup] = 150 / (m[sub]1[/sub] + 2)
t = sqrt((m[sub]1[/sub] + 2) / 150)
Finally, v=at:
u[sub]1[/sub] = (150 / (m[sub]1[/sub] + 2)) * sqrt((m[sub]1[/sub] + 2) / 150)
u[sub]1[/sub] = sqrt(150 / (m[sub]1[/sub] + 2))
Substituting into the earlier formula (and change m[sub]1[/sub]->m and v[sub]2[/sub]->v):
v = 2m*sqrt(150 / (m + 2)) / (m + 0.16)
At this point I’m going to cheat and use Wolfram Alpha to find the maximum (yeah, I could do the derivative, but it’s painful). It reports a positive maximum at:
m = 0.884 kg (v = 12.21 m/s)
All right; that’s 31 oz, which is rather higher than the (apparent) actual optimal number. So there’s obviously other stuff going on, or my estimates are off. But the OP asked about a 1 oz and a 50 oz stick, and I think we can see from the physics that “somewhere in the middle” is the right answer. The empiricists can deal with any remaining factors of two :).