Will this arrow fly 4800 ft?

Question

Scenario: An elf ranger with proficiency in the longbow, 16 DEX and the Sharpshooter feat prepares to shoot an arrow from her bow. A halfling wizard is positioned about 30 feet in front of the ranger and has readied a reduction spell to be cast on the arrow as it flies over his head.

The details on a Longbow are (PHB 149):

Longbow - 50 gp - 1d8 piercing - 2 lb. - Ammunition (range 150/600), heavy, two-handed

When the arrow flies over the halfling's head he casts the reduction spell and per the rules on Enlarge/Reduce (PHB 237):

The target’s size is halved in all dimensions, and its weight is reduced to one-eighth of normal.

Question: Will this arrow travel 4800 ft?

The answer may depend on whether the velocity of the arrow remains constant or the momentum of the arrow remains constant, and probably other factors as well (?).

Velocity: If the arrow maintains it's current velocity it will continue to travel at the same speed and will lose momentum. Momentum = mass x velocity (p=mv), so if the speed stays the same and the mass is reduced the momentum is also greatly reduced. In this situation, assuming the arrow weighs 20 grams and is traveling at 60 m/s it would look like:

original: p=.02 x 60, p=1.2 after reduction: p=.0025 x 60, p=.15

Momentum: If the arrow maintains its current momentum its speed will increase dramatically after it is reduced in size. Again, momentum is related to both velocity and mass. With the same numbers as above, keeping the momentum constant and changing the mass to 1/8 we would have:

original: p=.02 x 60, p=1.2 after reduction: 1.2=.0025 x v, v=480 m/s

Because the mass is reduced to 1/8 the original, the speed increases by a factor of 8. The reduced arrow is now traveling at 480 meters per second which is faster than some bullets (and also slower than many, it's on the slow end). The weight of the arrow is also about that of a small bullet. For context 480 m/s = 1575 fps = 1074 mph.

If the momentum does remain constant does this mean the mini-arrow will now travel 8x farther than the arrow at its natural size? Or does the loss in mass somehow counter the increase in speed? Can this elf/halfling duo shoot arrows up to 4800 ft in this way and essentially become a long range sniper?

The Enlarge/Reduce spell states:

The target’s weapons also shrink to match its new size. While these weapons are reduced, the target’s attacks with them deal 1d4 less damage (this can’t reduce the damage below 1).

While this may offer some guidance as to the amount of damage dealt, it does not address this particular situation. I'm looking for answers based in real world physics as well as any D&D based RAW or RAI that may provide some insight.

Answer 1

The rules for D&D do not simulate velocity or momentum

First of all, we always need to remember that the D&D rules are not a physics simulation, and trying to treat it like a physics simulation isn't going to work very well. In particular, the rules don't even attempt to model the concepts of velocity or momentum. Given this, we can only apply the "spells do what they say" principle. Enlarge/reduce doesn't say it changes an object's velocity, so the object's velocity does not change.

Physics can't tell you what happens when you violate the laws of physics

Obviously real-world physics doesn't specify what happens when an object's mass changes, because it is impossible to create or destroy matter. So going purely based on real-world physics, there is no way to meaningfully answer this. Any possible answer will violate the laws of physics in some way, since you're starting with a violation of the laws of physics (conservation of mass).

Strictly speaking, any violation of the laws of physics is just as bad as any other. However, to my mind, the least nonsensical answer is achieved by applying Newton's 2nd Law, which is that acceleration equals applied force divided by mass. We don't know what value to substitute for mass, but clearly the reduce spell does not apply any force to the object, so as long as the mass is non-zero (which it is), the acceleration is zero, which means the arrow's velocity does not change. This is equivalent to assuming that the arrow splits into two pieces, one carrying 1/8 of its original mass and the other carrying the other 7/8, and then the larger piece ceases to exist, leaving the smaller piece with the original velocity.

You probably can't hit an arrow with a spell in-flight anyway

The only way to hit an arrow in flight with a spell would be to ready an action to release the spell when the arrow flies by. However, if the trigger for your reaction is when the arrow is used to attack, your reaction only triggers after the attack is complete, i.e. after the arrow has already hit (or missed) its target. Essentially, the arrow is too fast.

Answer 2

Adding science to most parts of D&D is gonna break it. There aren't any rules saying it doesn't work the way you would like it to. However, there are no rules saying that it does.

5e is not meant to be realistic and doesn't have in depth rules for what your wanting to do because it Isn't meant for that.

If Enlarge/Reduce increased the range that the affected ammunition could travel, it would specifically say so.

The rule for ranged weapons is that an attack can be made at their first range increment normally, and with their second at disadvantage.

Although Sharpshooter allows you to make attacks without disadvantage, the Reduce option does not increase the range no matter what real-world logic you put into it.

Since D&D 5e gives power to the DM to make rulings, it is best to ask your DM.

Talk to the DM

In the end, this isn't supported by the rules but you put time toward it and find this kind of stuff fun. I don't see anything wrong with this being the entire goal for a campaign. Finding the knowledge in-character, practicing and testing to make this work, and then finally getting your shot at glory.

In my own personal opinion, I wouldn't allow it just due to it being against the nature of it being a role-playing game. Unless in a specific world and scenario, characters wouldn't have access to this knowledge. I am more than ok with "rule of cool", but D&D isn't meant for real-world logic, plain and simple.