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.

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    \$\begingroup\$ Mixing real world physics with the rules of a game is very messy and opinion based business. There is no correct answer here because the two simply don't fit together any more than Monopoly does with real life property law. \$\endgroup\$
    – kviiri
    Jan 30 '19 at 7:20
  • \$\begingroup\$ How is this opinion based? I'm specifically asking for a physics based answer. Physics is not opinion. Whether people agree that physics should be used in dnd or not is not in the scope of the question. That is something the answer posters have decided to address on their own. \$\endgroup\$
    – lightcat
    Jan 30 '19 at 8:46
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    \$\begingroup\$ You're specifically asking for a physics-based answer in a world where physics doesn't work because of magic. How magic works is up to opinion. \$\endgroup\$
    – kviiri
    Jan 30 '19 at 10:25
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    \$\begingroup\$ Note also questions about generic real-world topics are not on topic here when they do not draw upon RPG expertise. (See our on topic help, section “generic real world topics”.) This means “calculate some physics for me” would not be on topic for our site. \$\endgroup\$ Jan 30 '19 at 10:51

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.

  • \$\begingroup\$ @SevenSidedDie Those equations apply to systems with mass leaving or entering the system, which is not the case here. There is no "exhaust" as there would be for a rocket. Even if you try to apply that equation with an exhaust velocity of zero, the extra term drops out, leaving you with just F=ma and leading you to the same conclusion: no external force means no change in velocity. (In any case, like I said, you can't really apply physics to a situation that violates the conservation of mass.) \$\endgroup\$ Jan 31 '19 at 1:34
  • \$\begingroup\$ @SevenSidedDie There is no mass leaving the system. Ceasing to exist is not the same as leaving the system. Insisting on conservation of momentum while ignoring conservation of mass leads to a nonsensical result. You're effectively saying that when 7/8 of the object's mass disappears, that mass first "hands off" its momentum to the remaining 1/8 before disappearing. I stand by my claim that physics has no answer to what happens when you violate the laws of physics. \$\endgroup\$ Jan 31 '19 at 3:15
  • \$\begingroup\$ “Insisting on conservation of momentum while ignoring conservation of mass leads to a nonsensical result.” That's true. You're right, assuming p ^ ~p means I could conclude anything. In that case though, there's equally no support for saying in the answer that “physics says” velocity is conserved — it's just picking a different nonsensical assumption to make to get to that conclusion. I think your subsection then would be better simply saying what you just said: “Physics has no answer to what happens when you violate the laws of physics”. \$\endgroup\$ Jan 31 '19 at 3:33
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    \$\begingroup\$ @SevenSidedDie Fair enough. I'll edit that section it to better emphasis that "no answer" is technically the correct answer. \$\endgroup\$ Jan 31 '19 at 3:44

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.


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