Edits to “Statement”, “Solution”
en/2.3.10.md
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| ### Statement | |||
| − | $2.3.10.$ [Insert the problem statement] | ||
| − | |||
| − | ### 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 | |||
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| 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. <b>Bow strung too tightly (too stiff):</b> 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. <b>Bow strung too loosely (too weak):</b> 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. | |||
| @@ -22,7 +18,3 @@Solution | |||
| <b>How to choose the most suitable bow:</b> | |||
| 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: | |||
| $$E_{k (\max)} = \frac{1}{2} F_{\text{max}} l_{\text{max}}$$ | |||
| − | |||
| − | #### Answer | ||
| − | |||
| − | [Insert a concise answer or boxed result] | ||
| @@ -1,15 +1,11 @@ | |||
| ### Statement | ### Statement | ||
| $2.3.10.$ [Insert the problem statement] | |||
| ### 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? | $2.3.10.$ Why do bows that are strung too tightly or too loosely shoot poorly? How to choose the most suitable bow? | ||
| ### Solution | ### 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: | 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: | ||
| $$E_k = A_{\text{elastic}} = \frac{kl^2}{2}$$ | $$E_k = A_{\text{elastic}} = \frac{kl^2}{2}$$ | ||
| where $k$ is the stiffness of the bow and $l$ is the draw length. We can rewrite this formula as: | where $k$ is the stiffness of the bow and $l$ is the draw length. We can rewrite this formula as: | ||
| $$E_k = \frac{1}{2} (kl) \cdot l = \frac{1}{2} F l$$ | $$E_k = \frac{1}{2} (kl) \cdot l = \frac{1}{2} F l$$ | ||
| where $F = kl$ is the pulling force applied by the archer. | where $F = kl$ is the pulling force applied by the archer. | ||
| 1. <b>Bow strung too tightly (too stiff):</b> 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. | 1. <b>Bow strung too tightly (too stiff):</b> 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. <b>Bow strung too loosely (too weak):</b> 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. | 2. <b>Bow strung too loosely (too weak):</b> 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. | ||
| @@ -22,7 +18,3 @@Solution | |||
| <b>How to choose the most suitable bow:</b> | <b>How to choose the most suitable bow:</b> | ||
| 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: | 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: | ||
| $$E_{k (\max)} = \frac{1}{2} F_{\text{max}} l_{\text{max}}$$ | $$E_{k (\max)} = \frac{1}{2} F_{\text{max}} l_{\text{max}}$$ | ||
| #### Answer | |||
| [Insert a concise answer or boxed result] | |||