According to the laws of physics, a larger projectile WOULD have a greater drop off rate due to force due to gravity...
Hope you've got a tutor for physics class because you started being wrong right about here.
First point, gravity. Newton's law of gravitation. Gravity will affect both projectiles at the same rate because gravity accelerates all matter at an equal rate. Remember that story your science teacher told you about the apple and the cannonball being dropped from a tower and hitting the ground at the same time? I guess not.
You're right in that a heavier projectile requires a greater force to accelerate to the same velocity as a smaller one in the same amount of time, but you're assuming that the same force is being applied to the both projectiles. In reality (I'm talking the real world) you don't fire artillery shells using the same amount of gunpowder as a 9mm round, you scale everything up to be suitable for the projectile. In mechwarrior, the gunpowder is replaced with gauss coils, but for our intents and purposes they are the same. They both accelerate the projectile until it leaves the barrel.
Finally, drag due to air resistance. For objects moving in the same medium (the air), the values that affect the magnitude of air resistance are the surface area of the object and the velocity of the object. For simplicity's sake, we'll assume that projectiles are moving at approximately the same speed. The larger shell definitely has a greater surface area, so the force of air resistance on the large shell is greater than the force of air resistance on the small shell. HOWEVER, the larger shell has more mass than the smaller one. We had to compensate for that mass when designing the gun, but now that the projectile is in flight it works in our favor since greater forces have to be applied to make any appreciable change to its ballistic path. That means that not only will the heavier projectile be more resistant to slowing from air resistance, it will also be more accurate since it won't be blown off course as easily as the lighter projectile.
Here's an experiment you can do at home, wad up a sheet of paper into a ball. Now wad up some alluminum foil into a larger ball. The small lightweight paper ball is the smaller projectile. The larger, heavier aluminum ball is the heavier projectile. Throw both as hard as you can. See which one flies farther and more accurately.
also, take a look at these WWII naval guns.
16"/50 Caliber8"/55 CaliberThings worth noting, both guns have approximately the same muzzle velocity. The smaller 8 inch gun has a higher muzzle velocity at 853 m/s, while the heavier 16 inch gun has a slower velocity of 820 m/s when firing High Capacity ammunition. (think high explosive rather than armor piercing)
Also note the maximum ranges of both guns. The 8 inch gun has a maximum range of ~29 km, while the 16 inch gun has a maximum range of ~38 km.
In conclusion, from a physics standpoint, at long ranges the heavier projectiles should outperform their smaller cousins. If any weapon should suffer a damage dropoff, its the light weapons.
