How does the exchange of energy between an archer's arrow and their arm work? An archer cannot throw an arrow as far as they can shoot it, nor would its penetrating power be as great.
The obvious explanation is that they have traded strength for speed. A slow draw stores energy in the bow that is then released quickly.
The problem is that, when an arrow strikes its target it has a lot of energy and, as far as I know, penetrates further into the target than it would if used as a stabbing weapon in the hand of the archer.
How is this possible?  How can the use of a bow increase both range and penetrating power?  It seems paradoxical.
 A: As you said, "A slow draw stores energy in the bow that is then released quickly."
The speed of the string, and therefore the speed of the arrow, is much faster than the archer's arm can move.  The arrow's kinetic energy is proportional to the square of its speed.
Greater speed translates to greater range.
Greater energy translates to greater penetrating power.
Note that upon impact with a target, the arrow can exert a much greater force than the bow string exerts on the arrow, because the target may decelerate the arrow much faster than the string accelerates it.
A: Due to the physiology of the human arm, it's very difficult to throw an object much faster than 100mph. No matter what object is being thrown, it can't move faster than the hand that's holding it, so around 100mph is the upper speed limit of any thrown object.
A bow gets around this by using the elastic potential of the bow to store energy over an arbitrary period of time, rather than by trying to accelerate to maximum speed over the space of your arm's rotation. You can slowly and steadily pull back on the bow, storing more energy than you could have imparted with a throw. This allows arrows launched from a bow to reach 150mph or more. Since kinetic energy is proportional to the square of velocity, increasing speed by 50% more than doubles the amount of energy carried.
A: When an archer pulls back on a bow, it is the bow that bends. The energy that is able to be stored and released depends a lot on by the elasticity and tensile strength of the bow, and also the size of the bow, as that determines how much advantage you can get from leverage.
Leverage I would say is one of the main issues here.
Due to the length of the string, the archer has leverage over the ends of the bow, and is able to convert more of their own strength into the tensile material and therefore the propulsion of the arrow than they would by just throwing the arrow.
I'm not sure how to word it right, but I think there is also something to do around the concept that more strength can be used when there is more resistance/potential for inertia. Consider how it is easier to throw a palm-sized rock further than a small pebble, or a javelin further than an arrow

https://science.howstuffworks.com/crossbow2.htm
A: These answers are missing the point.
When you throw an arrow, most of the kinetic energy goes into your arm. When you draw it back with a bow, most of it goes to the bow and almost none goes to your arm.
A truck has way more energy than a baseball, but it's still better for a human to throw a baseball than hit it with a truck.
