In what scenarios would adding more mass to an object being thrown allow for more distance with the same force?
Your problem is that you are assuming a constant force. This is difficult to deliver. Since $F=ma$, in order to maintain a constant force as the mass decreases, the acceleration has to increase. For small masses, your arm can't accelerate fast enough to generate the required force.
It's easy to push on a baseball with 10N of force. It's very difficult to push on a pea with 10N. You may be pushing hard to throw the nerf dart, but most of that effort is going into accelerating your arm. By adding mass, your limited acceleration is able to apply a larger force to the dart.
Two initial observations :
- Are the darts launched with the same speed or the same kinetic energy?
When you specify that you use the same 'force' to launch both darts, it is not clear what you intend. Using the same force in Newtons over the same distance should result in the two darts acquiring the same kinetic energy but different speeds. However, as BowlofRed points out, your ability to launch objects is limited by the mass of your arm. If the mass of the arm is much larger than the mass of the dart, very little KE is imparted to the dart. All light-weight objects are launched with approximately the same speed, which is limited by size and strength of the arm and the technique used. See Physics of Overarm Throwing.
- The effect of gravity is a complicating factor here, so let us ignore it.
If there was gravity but no air resistance, the object launched with the greater speed would travel further. The range of a projectile without drag is proportional to $v^2$ and is independent of mass. If there is both gravity and air resistance the situation becomes more complex, except when the objects are launched close to straight up or close to horizontal. Darts are usually launched close to horizontal.
With Air Resistance (Drag) but no Gravity
For the same shape and size of object, travelling at the same speed, the drag force $F$ caused by air resistance is the same. It depends on cross-sectional area but not mass. For thrown objects, drag force is approximately proportional to speed through the air : $F=-kv$.
The equation of motion is $a=F/m$ so that $\frac{dv}{dt}=-(k/m)v$ which has the solution
$v(t)=v_0e^{-bt}$
where $b=k/m$. Integrating again we get
$x(t)=(v_0/b)(1-e^{-bt})$.
The distance travelled after a long long time is
$x(\infty)=v_0/b=mv_0/k$.
Which dart travels further depends on the initial momentum $mv_0$.
- Launch with the same KE
Here $\frac12mV^2=\frac12Mv^2$ (the lighter dart has higher velocity) so $V/v=\sqrt{M/m}$. The ratio of distances is
$x_M/x_m=Mv/mV=V/v=\sqrt{M/m}$.
The heavier dart travels further, despite the fact that the lighter dart started with higher speed.
- Launch with the same Speed
The above result indicates that the heavier dart will travel further still, because the lighter dart starts with less KE.
Here $V=v$ so the ratio of distances is
$x_M/x_m=Mv/mV=M/m$.
The heavier dart travels further still than when the darts have the same KE.
Conclusion : In both scenarios the heavier dart travels further.