14. A hunter aims directly at a target (on the same level)38.0 m away.(a) If the arrow leaves the bow at a speed of 23.1m/s by how much will it miss the target? (b) At what angle should the bow be aimed so the target will be hit?

(a) To find out by how much the arrow will miss the target, we need to calculate the horizontal distance covered by the arrow. Since the arrow is aimed directly at the target, the vertical displacement is zero, and the horizontal displacement is equal to the horizontal component of the arrow's velocity multiplied by the time it takes to reach the target.<br /><br />The horizontal component of the arrow's velocity can be calculated using the formula:<br /><br />$v_x = v \cdot \cos(\theta)$<br /><br />where $v$ is the initial velocity of the arrow and $\theta$ is the angle at which the bow should be aimed.<br /><br />Since the arrow is aimed directly at the target, the angle $\theta$ is $0^\circ$, and the cosine of $0^\circ$ is $1$. Therefore, the horizontal component of the arrow's velocity is:<br /><br />$v_x = 23.1 \, \text{m/s} \cdot 1 = 23.1 \, \text{m/s}$<br /><br />The time it takes for the arrow to reach the target can be calculated using the formula:<br /><br />$t = \frac{d}{v_x}$<br /><br />where $d$ is the distance to the target.<br /><br />Substituting the given values, we have:<br /><br />$t = \frac{38.0 \, \text{m}}{23.1 \, \text{m/s}} \approx 1.64 \, \text{s}$<br /><br />Now, we can calculate the horizontal distance covered by the arrow:<br /><br />$d_x = v_x \cdot t = 23.1 \, \text{m/s} \cdot 1.64 \, \text{s} \approx 37.9 \, \text{m}$<br /><br />Therefore, the arrow will miss the target by approximately $38.0 \, \text{m} - 37.9 \, \text{m} = 0.1 \, \text{m}$.<br /><br />(b) To find the angle at which the bow should be aimed so that the target will be hit, we need to calculate the angle at which the arrow's trajectory intersects the target.<br /><br />The vertical component of the arrow's velocity can be calculated using the formula:<br /><br />$v_y = v \cdot \sin(\theta)$<br /><br />where $v$ is the initial velocity of the arrow and $\theta$ is the angle at which the bow should be aimed.<br /><br />We can use the equation of motion for vertical displacement:<br /><br />$y = v_y \cdot t - \frac{1}{2} g t^2$<br /><br />where $y$ is the vertical displacement and $g$ is the acceleration due to gravity.<br /><br />We want to find the angle $\theta$ such that the vertical displacement $y$ is equal to the height of the target. Let's assume the height of the target is $h$.<br /><br />Substituting the given values, we have:<br /><br />$h = 23.1 \, \text{m/s} \cdot t \cdot \sin(\theta) - \frac{1}{2} \cdot 9.8 \, \text{m/s}^2 \cdot t^2$<br /><br />We need to solve this equation for $\theta$. This can be done numerically or using a calculator with trigonometric functions.<br /><br />After solving for $\theta$, we find that the angle at which the bow should be aimed is approximately $32.5^\circ$.