Someone has a magic power that allows them to compress and expand objects.
Mass remains the same to preserve the law of conservation of mass.
Two scenarios:
What would happen if this person shrunk a 19m x 5m ferro magnetic metal slap to the size of a .50 BMG round and fired it from a railgun?
What would happen if this person compressed or expanded bullets fired at them? Which would be more effective at protecting them?
First, a physics note. If you compress or expand something in this way, its linear momentum and kinetic energy won't change (as you've rightly gathered). But the moment of inertia depends on how spread out the mass is. So if a spinning object is expanded, it will spin slower; if it's compressed, it will spin faster. The classic demo of this is a twirling figure-skater, who speeds up when she pulls her arms close to her body. This could violate conservation of energy depending on how much energy it takes to compress things; if spinning objects "resist" being compressed (and thus take more energy), conservation won't be a problem.
For non-combat uses, the scientific potential is tremendous. Forget microscopes: just scale up your samples until you can see the details you want. Want a more powerful phone? Take a high-end processor and scale it down. Depending how far this expansion and compression can go, the possibilities are endless!
But of course, the military would probably get their hands on it first.
As far as combat usage, your metal slab could be enormously effective, but also enormously difficult to use. Let's say the slab is one centimeter thick. Wolfram|Alpha says that steel alloy is about 7.859 g/cm³; your 19x5x0.01 slab would weigh almost 7.5 metric tons. The heaviest .50 BMG bullet I can find is only 52 grams. So your railgun will need quite a lot of power to even push it out the barrel.
It would probably be more effective to fire a larger object at the beginning: a cannonball, for instance. Use a proper artillery cannon to get it moving. Then compress it into a bullet in midair. It'll still have all the kinetic energy of a flying cannonball, but now that energy is focused on an area smaller than a quarter; that will tear through pretty much anything.
If your enemies return fire, you'll want to do the opposite. Bullets hurt because they're so small, so their force is focused onto a small area. Scale them up a few orders of magnitude, and you're effectively being hit with a styrofoam ball; it might leave a bruise if it's moving fast enough, but it won't kill you. (Plus, as John pointed out, air resistance will stop it dead fairly quickly.)
Confirming this experimentally is left as an exercise for the reader.
This is a pretty strong superpower -
Bullets are no problem as long as the power works fast enough to expand them enough in real time. In addition to expanding them into styrofoam mid flight, you can expand them before they're fired so that they get stuck in the barrel and explode/disable the gun. Or shrink them within the clip so that they don't fire at all. Or you can disable the gun by messing up the barrel, trigger etc.
You can manipulate air pressure and create strong winds, hurricanes, tornadoes, potentially rains and what not.
You can freeze or boil air, water etc. You can freeze someone's blood in their hearts or explode one's blood vessels or shrink heart/brain until they die.
You can freeze the water in the air and drop on / bombard the opponents with ice-spears or giant hail-cannon-balls.
You can freeze water into a solid walls of ice around you to defend against everything from arrows to machine gun fire.
You can shrink the load bearing beams/pillars of buildings and destroy them.
You can manipulate air densities to to make yourself blurry or difficult to see/target.
You can contain explosions (grenades, explosive rounds) by compressing the air (around the bomb) faster than it can expand.
You can dismantle tanks and any vehicles by shrinking/expanding their guns, engines, tracks, tires etc.
You can turn any containers or empty bullet casings into deadly projectiles/bombs by violently expanding the air inside them (preferably after first filing them with compressed air).
You can probably turn a frisbee into a deadly weapon by compressing it into flat very sharp edged disk right before it hits your unfortunate opponent's neck...
The possibilities are endless depending on the range and potency of this power.
Oh and if you were really strong and kept on compressing stuff, you could also end up creating black holes...
As far as offensive capabilities go, depending on the range of your ability you have huge possibilities. Pick a particularly important object in your opponents body, lets say, the skull, compress it down without any concerns about its contents, and you just instantly killed your opponent in the most gruesome looking way imaginable. If you can't target one part of an object, then perhaps expand whatever they had for lunch, or just shrink any threats to insignificant sizes.
However for defending against projectiles, it may not be as useful as you'd think. Conserving both mass and velocity means that the force behind the object is also conserved, so our best options are to change the magnitude or direction of the force in some other imaginative way. Expanding an object into a shield is a possibility, but keep in mind that this larger object has the same mass and therefore a smaller density. Expand your block of titanium large enough and all you have is the worlds shiniest piece of paper.
Perhaps a more useful ability would be to alter the shape of the projectile, that way you can add instability to the bullets drag and force it to miss its target. If it is possible to target part of an object, then a skilled user, or multiple users working closely, could expand one side of the object and compress the other.
Expand the bullet in two dimensions, so that you suddenly have a giant slab of metal flying at you, but with a very low density, akin to metal foam or iron wool. Once bullet bill has slowed down enough, you can compress him back down to regular size, and allow the now slowly moving bullet to fall harmlessly to the ground. Calculate the trajectory fast enough and you can pluck the once-deadly bullet out of the air for extra style points.
Uh oh, there's a bullet directly in-front of my face, why did they have to make these things so darn fast‽ At this point you have one fall-back plan, compress the projectile as small as you possibly can, if an unimaginably small needle hits you with the force of a speeding bullet, it will likely penetrate your entire body with a hole too negligibly small to be concerned of. If you find yourself faced with a hailstorm of bullets, this is also a possible option, compress every object in-front of you and hope that none of these needles get stopped by bone or sever a particularly important neuron.
If you compress the bullets but your powers keep their mass and velocity the same (= preservation of momentum), you haven't helped yourself a lot. The bullet will still hit you and will still punch a hole through you; just that the hole isn't a 9mm hole but a 1mm hole (or thinner). Actually, if you make a regular handgun bullet smaller, then the likelihood that it becomes a 'clean' through-and-through (instead of getting stuck in a bone) is a lot greater. The area of impact is smaller, so the penetration capability becomes greater. Kind of like, if you hit a piece of wood with a hammer, it might get some dents. But if you hit a nail with the hammer and the same amount of force, it gets driven into the wood very easily.
So, basically, compressing helps a bit with the size and severity of the wound, but doesn't eradicate the problem unless you go down to a micro- or nanometer level (so that the hole becomes so small that you don't realize it anymore). Also, the question remains what will happen with the bullet afterwards -- does the magic wear off, or is this a new way of creating super-dense materials?
If you expand the bullets, you've got more of a chance of 'stopping' it. Expand it from 9mm to 90cm, and it suddenly gets so much air resistance that it will drop to the ground within 2 or 3 meters, or get completely blown off-trajectory. Since it still weighs as much as a single bullet but has the size of a bean bag, you could compare it to getting shot at with a large piece of styrofoam. You might get a bruise, but nonlethal.
In both cases though, you need to be fast enough to work your magic! If there is any kind casting time involved -- the bullet will hit you before you've finished the spell. If you need to focus your attention on the bullet after it leaves the muzzle, you are SOL - human reaction time simply can't handle such speeds. The only way to actually stop bullets with that kind of magic would be creating compression / expansion fields where anything that enters the field becomes compressed / modified. And that brings a whole slew of different problems with it! Or you need 'program' your field magic to only work on projectiles above a certain speed, or something like that...
You could expand the space between yourself and the shooter. That way, it looks like you are 5 meters apart, but the bullet actually has to travel 5 miles to bridge the distance. Should it arrive at the target at all, it will have lost most of its momentum and be completely off-target.
Also, you could compress the space inside the pistol / gun's muzzle. This will definitely mess up the shooter's aim because the bullet leaves the gun much too early. The powder's explosion capabilities might not all become transferred to the bullet, so it's not that fast anymore.
You could compress the air between you and the bullet so that it becomes as dense as water and robs it of much of its velocity.
With compression/expansion, you automatically get awesome refridgeration / heating powers. Expanding a substance but keeping its mass the same cools the substance (the principle on which fridges work). On the other hand, compressing a substance radically heats it up (huge example: hydrogen clouds in space are slowly drawn together by gravity and become tighter and tighter until one day they're dense enough to ignite into a new star).
Depending on how long-lasting the effects of compression/expansion are, you also have a way of manufacturing super-dense materials that are an awesome bullet shield although they are super-thin. Doesn't do much for the weight (still weighs as much as a 6-inch slab of steel), but at least it looks nicer than a tank.
Such super-dense materials should also have some other interesting physical properties, depending on how you achieve the super-density. Either you decrease the space betwen individual atoms, or you shrink the atoms themselves, too -- any way, you will be messing around with some of the most basic forces in the universe, and as a result I bet you will be getting all new and exciting physics out of it!
to Question 2, you could expand a bullet until it has the density of styrofoam, in which case it can't hurt you and air resistance stops it dead in a few inches. If mass is maintained expanding it drops its density. to clarify a 9mm bullet would have to be expanded until its about 10 inches cubed (roughly the size of a small coffee cup.) to reach this density, at which point it would either be stopped by air resistance or bounce harmlessly off of you. shrinking a bullet would not make it less dangerous. The hole it makes would just get smaller and deeper.
Why use the ability on someone's bullets when you can use it on the gun itself. Shrink the barrel of someone's gun by just 1%, and the bullet will be wedged in there tight enough to turn that gun into a pipe bomb with a trigger.
As for the railgun idea, as the way you describe it, you would be firing an ultra-dense massive slug, which would be virtually impossible to accelerate to speeds that can fly. Accelerating a few tons of steel to supersonic speeds takes a lot of energy... its why fighter jets have such insanely large engines strapped to them (or perhaps more accurately, fighter jets are made of an engine with a cockpit and landing gear strapped to them)
science-fiction? Magic powers are typically not a part of science fiction (Star Wars not withstanding). $\endgroup$