Solution

Statement

$2.3.10.$ Why do bows that are strung too tightly or too loosely shoot poorly? How to choose the most suitable bow?

Solution

A bow shoots "well" when it imparts the maximum kinetic energy (and thus maximum velocity) to the arrow. The kinetic energy $E_k$ of the arrow is equal to the work done by the elastic force of the bowstring:
$$E_k = A_{\text{elastic}} = \frac{kl^2}{2}$$
where $k$ is the stiffness of the bow and $l$ is the draw length. We can rewrite this formula as:
$$E_k = \frac{1}{2} (kl) \cdot l = \frac{1}{2} F l$$
where $F = kl$ is the pulling force applied by the archer.

1. Bow strung too tightly (too stiff): The archer applies their maximum physical force $F_{\text{max}}$, but due to the high stiffness, they can only draw the bowstring a very short distance $l$. The stored energy $E_k = \frac{1}{2} F_{\text{max}} l$ will be low because $l$ is small.
2. Bow strung too loosely (too weak): The archer can easily draw the bowstring to their maximum anatomical arm span $l_{\text{max}}$, but the force $F$ required to hold it is much less than their maximum capability. The stored energy $E_k = \frac{1}{2} F l_{\text{max}}$ will be low because $F$ is small.

How to choose the most suitable bow:
The most suitable bow is one whose stiffness $k$ allows the archer to reach their maximum pulling force $F_{\text{max}}$ exactly at their maximum anatomical draw length $l_{\text{max}}$. In this case, the kinetic energy transferred to the arrow will be maximized:
$$E_{k (\max)} = \frac{1}{2} F_{\text{max}} l_{\text{max}}$$