Let's break down the problem step by step. We are given: - A baseball is hit directly over second base with a velocity of **85 mph** (let's call this vector \(\vec{v}_b\)) - A wind of **24 mph** is blowing from third base to home (let's call this vector \(\vec{v}_w\)) We are to find: 1. The **resultant speed** of the baseball. 2. The **direction** of the resultant velocity (relative to the original direction in which the ball is hit). We'll assume the field is a square: - Let home be at (0, 0), first base at (a, 0), second base at (a, a), and third base at (0, a). - The ball is hit from (0, 0) towards (a, a), i.e., along the line \(y = x\). - Wind blows from third base (0, a) to home (0, 0), i.e., along the negative \(y\)-axis. ### Step 1: **Express the velocities as vectors** #### Ball’s velocity (\(\vec{v}_b\)) - Direction: from home to second base (along \(y = x\)) - Unit vector in this direction: \(\left<\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right>\) - So, \(\vec{v}_b = 85 \left<\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right> = 85 \left<0.7071, 0.7071\right> \approx \left<60.10, 60.10\right>\) mph #### Wind’s velocity (\(\vec{v}_w\)) - Direction: from third base to home (along the negative \(y\)-axis) - So, \(\vec{v}_w = 24 \left<0, -1\right> = \left<0, -24\right>\) mph ### Step 2: **Find the resultant velocity** \[ \vec{v}_r = \vec{v}_b + \vec{v}_w = \left<60.10, 60.10\right> + \left<0, -24\right> = \left<60.10, 36.10\right> \] ### Step 3: **Find the magnitude (speed) of the resultant vector** \[ |\vec{v}_r| = \sqrt{(60.10)^2 + (36.10)^2} = \sqrt{3612.01 + 1303.21} = \sqrt{4915.22} \approx 70.13\ \text{mph} \] ### Step 4: **Find the direction (angle) relative to the original direction** - The original direction (home to second base) was at \(45^\circ\) to the x-axis. - The resultant makes an angle \(\theta\) with the x-axis, where: \[ \theta = \tan^{-1}\left(\frac{36.10}{60.10}\right) = \tan^{-1}(0.601) \approx 31^\circ \] - The original direction was \(45^\circ\), so the angle between the resultant and the original is: \[ \alpha = 45^\circ - 31^\circ = 14^\circ \] So, the resultant is **14° below** the original direction. --- ## **Final Answers:** ### 1. **Resultant Speed:** \[ \boxed{70.13\ \text{mph}} \] ### 2. **Direction:** \[ \boxed{14^\circ} \text{ below the original direction (home to second base)} \] or \[ \boxed{31^\circ} \text{ above the x-axis (home to first base direction)} \] --- **If you need the solution for the second part (wind = 22 mph), just let me know!**