I think the long bows were mostly for long range, rapid fire. When first introduced, long bow troops stood out of the range of fire from an enemy army and showered down a rain of destruction on them from the sky. This could decimate an army well before they were within range to attack.
Shorter bows were more suited for calvary units who could ride in fast, shoot rapid fire and fairly accurately since they were at close range, then retreat quickly. These tended to be made from laminations of various material to give them greater strength.
Crossbows were mostly for unskilled soldiers as they were easier to aim and shoot, but they lacked the rapid fire capability and strength of standard bows and long bows. It took great skill to shoot long bows and shorter bows, and those troops were harder to train. Crossbows were for lesser skilled troops and not as effective in general.
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