The question should be more general than how much energy does it take to kill and elk. After all people have killed polar bears and tigers with 22lr so a 22lr should be able to kill an elk. The question you should ask is what is the recommended cartridges for elk hunting.
You don't get many opportunities at an elk so you should use a cartridge that maximizes you chances for success. You need to have penetration from what every angle you happen to see him and out to 300 yards or so. You also want something that will drop the elk relatively close to where its shot. Having it run 5 miles or more and dying in the brush is not a good thing.
The generally accepted best cartridge for elk is the 338 mag. It will do all that you need for elk. The general big game cartridges - 270, 308, 7mm mag, 3006, 300 win will do ok though it would be best if you use controlled expansion bullets for better penetration.
don't tell that to the 10's of thousands of elk that have been killed with less.
elk aren't superior ironsided steel boned super animals. they are just really big deer. I've killed them with a variety of cartridges and calibers and bullet weights. from a 90 gr bullet moving fast to a 180 grain bullet moving slow. from a 25 caliber to a .30 cal and you know what? they all killed elk when hit in the right place, and they all died within feet of where I shot.
this question has been asked and asked and asked and asked.
if you're worried about energy, some states have a requirement, I don't remember what it is but something like 800 ft/lbs at 100 yards. and a minimum of 24 caliber. some guys use the 1000 ft lbs of energy at moment of impact as a reference. if the bullet doesn't go where it needs to, its a moot point. shoot something big enough to take down a big deer, and hit them in the vitals. nature will take care of the rest.
