Would melee weapons become more useful than guns, or would guns still be better?
For reference: in terms of speed, the enhanced humans can outpace bullets by fast-walking.
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Bullets aren't the only kind of weapon out there, and there are plenty of weapons designed explicitly to be hard to miss with.
Volume-of-fire would be king. Contrary to many fictional depictions, a gatling gun sweeping across your position pretty much can't fail to hit you. thousands of bullets per minute can't really be dodged. Even if they're apparently going at tennis-ball speeds.
In a similar vein, Shotguns and similar scatter-guns are pretty well impossible to dodge even if you can treat the pellets as low subsonic. It doesn't really matter if you can get out of the way of the worst of it, being crippled by the parts you didn't dodge leaves you open to a coup-de-grace.
I would also expect a general military shift towards weapons that can't be easily dodged.
Flame-throwers and incendiary weapons Grenade-launchers and similar explosive-launching weaponry such as the XM-25 which shoots airbursting grenades to hit targets in cover.
At the tactical level, use of mines and hazards to funnel enemies into a kill-zone is already a well understood practice. Being able to dodge bullets doesn't work in a confined space. If you can't get out of the way then there's nowhere to dodge to.
I would also expect an emphasis on supersonic ammunition. If you can't hear the gun before the bullet hits you, then you won't know you need to dodge in the first place.
Being able to dodge bullets is one of those things that only works if you're under ideal circumstances, and warfare is all about making sure that your opponent never ever gets their ideal circumstances.
Who Needs Guns?
A gun is a machine that uses chemical energy to accelerate a projectile to (usually) above the speed of sound. When this projectile hits a target, it transfers that energy to the target, damaging it.
In your world, a human can move at supersonic speeds. (Analysis for scenario one: A tennis ball serve is around 120 mph, about 20% faster than a baseball pitch. In a baseball pitch, the pitcher's hand is moving at the ball's exit velocity. A typical bullet is more than 120% the speed of sound, so your scaled human's pitching hand would be supersonic. Analysis for scenario two: Trivial.)
This means that a human can throw a rock with velocity comparable to a bullet, but with an order of magnitude more mass and thus more energy. (A baseball weighs about 145 g, a 45 caliber bullet weighs about 15 g.) Why would you use a gun to accelerate a projectile when you can just throw it?
[Side note: Similarly in close-quarter combat you probably don't need melee weapons, and man-scale armor is probably useless. The sheer amount of energy in a supersonic punch will obliterate a human target absent super strength to go along with super speed and nearly-infinite energy. Carrying a weapon would be overkill, and being even slightly encumbered could only reduce your ability to dodge similarly-fast opponents.]
The question, then, is whether it's worth throwing projectiles at the opponent. It's very cheap to try it, ranging from the cost of carrying a bag of projectiles down to (if there are pebbles around) zero. And your opponents can move fast enough to dodge, but they can't actually hear the projectile; they have to see each individual projectile to take action to avoid it. If you ever see the back of an opponent's head, chuck a rock at it and see it explode in a red mist. If you have three buddies, throw three rocks such that dodging the first puts you in the path of one of the others. If nothing else, why not try? Unless the sun catches it just right they might not notice...
Just because your people are a lot faster doesn't mean they think faster or sound travels faster. A lot of the same weapons would still be used - they would just go through more ammo. But in addition, melee weapons and machine pistols would become a lot more popular.
Tactics in combat would need to change. It would be difficult to shoot anyone who knew they were being shot at using slow-firing weapons, since they could reasonably keep dodging around to avoid being hit. But you could still surprise people, since they wouldn't hear the bullets coming. You could still spray bullets at opponents, since they couldn't see the bullets to know exactly where the bullets were. So your troops would try to drive enemies into positions where there was little maneuvering room. While the opponents could dodge faster, shooters could also aim and fire fully automatic weapons with ease.
But troops with this kind of speed would likely be able to rush enemy positions and be extremely close to enemies. Setting off explosives and shooting projectile weapons at extremely close quarters would be a recipe for disaster. Hand-held weapons would move at the speed of the people wielding them, so a rapier or katana would be a very effective weapon. But even the humble K-bar would be deadly to opponents at close range.
Mortars, RPG's and grenades might be big and slow enough to allow people to both dodge and deflect them mid-air. To counter this, you would need these weapons to be radar triggered by proximity.
While vehicles with powerful, fast-firing guns will still be useful, really important vehicles like tanks will likely be obsolete. A man carrying a large-enough satchel charge could run up to them and blow them up.
Different guns have different bullet speeds. BB guns are the slowest, comparable to paintball shots. Most guns nowadays however shoot bullets that come out of the barrel faster than sound.
If everyone can suddenly dodge bullets from a rifle, what will happen is that armies around the world as well as Americans will demand less obsolete guns and ammunition, and more high speed weapons and bullets from their suppliers. Sabots would also become more popular. Those can make bullets faster, enough that your people wouldn't be able to outrun them.
But if your guys can dodge sabot-accelerated bullets, then your world might switch to laser guns faster than the real world. Only Judas Priest's Painkiller would be able to outrun that.
Chances are the development of laser weapons would be ramped up. Youve got a few years before we figure it out, but once practical man-portable lasers are built, these super soldiers wouldn't be so super. They can move faster than bullets, but nobody is moving faster than the speed of light.
The simple answer is that somebody would build a better gun. For example, they would switch to a scattershot weapon that put out a wall of fire to large to easily avoid, or they would use a gun with a higher rate of fire.
Examples might include a superposed load gun, which is a real world weapons type in which multiple projectiles are launched from a single barrel. Such as the experimental Metal Storm system enter link description here or Gatling weapons.
Melee weapons might be given more emphasis as a backup, but it's likely that armies would deploy methods to make closing to melee range difficult. Such as using more fragmentation weapons, which would be hard to dodge.
If one can outrun bullets, their body would be so strong that they wouldn't need to. As others pointed out, faster bullets could solve this problem.
But, there are other weapons that could damage the bodies of the supersoldiers. The weapons could be chemical, radioactive, etc.
One could go the chemical route by finding what sort of acid, nerve agent, etc. can kill these supersoldiers. Then, one could use tanks to fire special projectiles which explode over large areas. Or they could use mines to spread the destructive chemical. When the supersoldier appears, an entire area surrounding them could be covered with the nerve agent causing the soldier to die.
Another weapon that could work, assuming these soldiers need a lot of oxygen to move fast, is some sort of thermobaric bomb. As the supersoldier suffocates because of the lack of oxygen due to the bomb, someone slower could move in for the kill with a projectile weapon.
Some strong source of gamma rays cold cause serious radiation sickness to these soldiers. If one could make a weapon with a lot of gamma radioactive material, in principle, the gamma rays could be focused on the enemy with a bunch of Laue lens. This would be impossible to dodge since gamma rays are invisible.