To clear up a few misconceptions:
Modern compound bows are more efficient than medieval longbows so you will have to determine how much heavier a longbow must be in order to match the kinetic energy of a lighter compound bow. Most standard bow range from 25-45 lbs, while longbows 100-130, some up to 185 lbs.
@Lurker
Modern bows vary greatly based on manufacturing company, design of the limbs, cam configuration, etc. There are plenty of compound bows that shoot with kinetic energies much greater than your listed 68 lbs. The most powerful 70# compound bow shoots a 350 grain arrow at a KE of 106 ft-lbs. Best estimate for the weight of longbow arrows are 60-115 grams, which is about 975-1750 grains approximately with 33-34'' length, while the weight of compound bow arrows range from 350-550 grains assuming a 125 grain tip and 31-32'' arrow length. The heavier the arrow, the higher the Kinetic energy when launched from the same bow.
I can give a comparison of the power of a 70# compound bow versus a heavy English warbow and determine its penetration power if you want. I don't have much experience with longbows, but I own a 2014 PSE DNA sp, one of the fastest compound bows on the market, of 70# and its IBO speed is set at 345 FPS. So it shoots a 350 grain arrow at 345 fps, giving 92.5 ft-lbs of KE. However, the heavier the arrow, the higher the kinetic energy because heavier arrows allow for better transfer of energy from the string to the arrow. I shoot 548 grain Easton FMJ arrows at 300 fps at 30'' draw, which gives my 109.5 ft-lbs of KE. That's almost as much KE as the 112 ft-lbs Ancient mentioned shot from a 150 lb warbow. However, due to FMJ arrows having much thinner diameter shafts (9/32 inch versus 3/8 or 1/2 inch for longbow arrows), the penetration of the compound bow's projectiles will be MASSIVELY superior.
Oh yeah, the compound bow that I use delivers superior power than some modern crossbows as well. It has similar kinetic energies to the fastest TenPoint crossbows on the market, and TenPoint makes some of the best crossbows there is. Then again, TenPoint is renowned for the durability, reliability, warranty, and customer service that come with their weapons, not speed.
There are also other variables to consider. One important one is the extensive training longbowmen had back in the day. They were able to load and shoot at much higher rates than modern compound bows can due to simplicity and training, said to be at 15 arrows/min but contradicted by longbow club members today who are able to loose 25-30 arrows/min. That is a rather insane rate of fire for an archer, enough to constantly pound enemy cavalry with enough blunt force to knock them to the side. Another is the use of bodkin points, which were designed to defeat PLATE, not chainmail. Bodkin points are thinner than field tips so had better penetration qualities, but field tips are still pretty sharp and can penetrate better than most broadheads. The downside is that after the bodkin tip goes through the armor, the shaft will fail to do so because of how thick warbow arrow shafts are.
It is also true that modern compound bows are ridiculously accurate though. With modern sights and peep sights along with a mechanical release aid, a modern hunter can easily hit a knight from long range. Longbows had a maximum effective range of 300 yards, while modern compound bows can shoot up to 550 yards, giving the DNA sp a substantial range advantage. Of course, accuracy is low at these distances.
Overall, I believe that the top end compound bows today like the PSE DNA sp have superior penetration than 150 lb warbows just because of similar KE and much less cross-sectional area of modern arrows. Seeing as longbows can penetrate plate armor at 20 yards, a modern bow hunter could easily do so. However, the knight is also wearing padded gambeson underneath the plate, and gambeson itself can sufficiently nullify the arrows of a lesser longbow. The fight could go either way, depending on what the knight is wearing under the plate and the quality of the plate armor.
