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How would lower gravity, let’s say around ½ of Earth’s, change the efficacy of different Bronze Age projectile weapons? How would the effective range, accuracy, and penetrating power of bows, javelins, atlatls, and slings be influenced?
Does lower gravity favor any of these systems over the others? How might these weapons be modified from their Earth counterparts to be more effective in a lower gravity? For example, would projectiles be made more massive?
As @theRiley points out in comment, also note that the reduced gravity will likely mean your people have reduced muscles and therefore reduced strength. Because of this, projectiles which rely on the strength of the shooter, such as thrown rocks and spears, will not have the same range and momentum as they would if thrown by someone visiting there from Earth.
All other things being equal, the main effect of the gravity will be that the object does not fall back to the ground as soon and will instead have a higher range and possibly improved accuracy.
All gravity does is pull things toward the ground. In fact, if we ignore objects with different aerodynamic profiles for a moment, we can say that all objects fall to the ground at the same rate. It is a common sight in introductory physics to see people get all riled up when you tell them that a bullet shot out of a gun parallel to the ground will hit the ground at the same time as an object released at the exact same moment to fall to the ground.
A projectile will travel in an arc along a parabola. The higher the gravity, the greater the change of direction in the curve. So to project farther, you have to aim higher. This much is intuitive: if aiming a bow or a gun at a distant object, you actually have to aim above the object. In lower gravity, you would not have to aim quite as high. Your shot would be straighter for a target at the same distance, so the accuracy would be improved.
If going for distance, since the object does not fall to the ground as fast, it will keep moving away from you and will have a farther range.
All other things being equal, nothing should change significantly concerning penetrating power, at least not that I can think of.
More massive? Obviously yes. Since gravity is reduced, you can more easily project a greater mass. I could pick up a larger javelin to hurl. However, though I can pick up and throw a larger mass, the larger mass will still have a larger resistance to change in momentum. I can throw a larger rock or javelin, but since its speed will be lower we will not have a linear increase in momentum with larger objects (ie: just because the rock is twice as massive doesn't mean it will hit twice as hard, since I cannot throw it as fast as a lighter variant).
We see this same effect in modern guns. Some handgun owners think that their .45 caliber handgun is so powerful because shoots such big bullets. But it takes more energy to get those bigger bullets up to speed, so often the .45 caliber handgun bullets are being shot at slower speeds than smaller bullets. For some ammo, the total momentum (and thus the total impact force) of a .45 is only slightly higher than a smaller round like a 9mm.
Same thing with your more massive ammo: you can hurl larger shots and they will go further, but at a certain mass there will come a point when you are not increasing the actual momentum and therefore not increasing the impact force hitting your target. At what point these diminishing returns get in the way I'm not sure, but I doubt the size and mass of projectiles would increase by a lot. Probably no 1-foot-diameter sling stones for example.
Range: increased. Accuracy (given the same distance as a full-G target): increased. Ability to project higher mass: increased. Impact force from higher mass: increased but only marginally, with diminishing returns, just like we have on Earth.
The range of a projectile launched with initial velocity $v_0$ at an angle $\alpha$ is given by $r=$$v^2_0 sin(2\alpha) \over g $.
We see immediately that changing $g$ affect the range. In your case, the range would be doubled at any angle.
The damage delivered by the projectile is dependent from its initial kinetic energy, which is $K=1/2mv_0^2$. Here we see that gravity doesn't play a role.
Summarizing, by lowering gravity and keeping all other factors, such as projectile mass and initial velocity, the same, you only increase the range of the weapon. The rest will remain unchanged.
If you want to increase the damage, you have to increase the mass of the projectile and, since the energy that the launcher can transfer to it is fixed, accept a proportional reduction in range.
In the formula above drag is neglected, but the final deduction still holds.
