While it is my solemn KMC Avengers Duty to defend Hawkeye on this forum, I don't really know Hawkman's abilities (other than that he flies and has a mace) to the point where I can knowledgably comment here.
Just how strong/fast/durable is Hawkman?
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Strong enough to make the likes of Black Adam and Superman feel the hits from his mace, Nth metal harness increases his strength, and durability too (although he is no means arrowproof). His healing factor is quick enough to heal a completely burned arm in about a half a hour. He can achieve speed of sound pretty easily, and surpass it without effort too. He's probably faster then normal humans too when it comes to reaction speed.
No matter how good Hawkeye is with his bow, I'm not sure he'll be able to hit something that goes the speed of sound........ and if Hawkman get close to him, it's over. I'll take Hawkman 9/10.
OK, I've just finished reading the EXHAUSTIVE Hawkman mega-bio (which also reminded me why I don't read very much DC) over at monitorduty.
Current Hawkman certainly overmatches Clint physically, but that's hardly ever stopped him from taking people down before (She-Hulk, Abomination, etc), nor does he usually have trouble with fast fliers (other than Kree jetpacks).
Hawkman could draw this by just flying away, but if he's going on the offensive he won't be close to the speed of sound. Unless I'm mistaken, he seems to have a highish max-speed, but not hyper-reflexes like Superman or whomever. He's agile enough for some good dodging, but as usual Hawkeye doesn't need a direct hit to take him out.
The Antarctic Vibranium arrow he used to take out Alkhema would send a suddenly-normal-human Carter dropping to like a stone as his harness dissolves, but in fairness that's a prepped attack.
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...You are Number Six. Respect Popeye
Then what is? That's basically Hawkeye's only hope of winning this fight and he doesn't even have that. Hawkman's stronger, faster, has better equipment, has a healing factor, is a better fighter (Yes, that's right. Hawkeye has 1000 years worth of experience for fighting because he remebers all of his past lives), etc... Hawkman should win this fight 9/10.
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Bam Hawkeye draws him in Flasbang Arrow's him, sends in a sonic arrow to disrupt his stability, electric arrow to stun and Nuke arrow to put hawkman into a nice state of "healing" for a day or two!
Well to be fair, Hawkman can see the freakin Flash when hes vibrating (Idenity Crisis #2) which will be a huge advantage in this fight (the seeing I mean).
But, Hawkeye is no pushover. I mean like Cap he earns the right to be on the same team with powerhouse heros like Iron Man and Thor.