Let's break down the problem into three steps with explanations and diagrams as requested. --- ### **Step 1: Understanding the Problem and Drawing the Diagram** **Problem Statement (Summary):** - A quarterback is preparing to throw a pass to a receiver. - The receiver is 15 yards down the field and 20 yards to the right of the quarterback. - The receiver is running at 9 yards/s at an angle of 45° to the right of straight down the field. - We need to express the initial velocity vector of the football **relative to the field**. **Diagram:** Here's a simplified version of the scenario: ``` y (downfield) ^ | | Receiver | * | /| | / | 45° | / | |/___|________> x (right) Quarterback ``` - Quarterback at the origin (0, 0). - Receiver at (20, 15) in yards (x, y coordinates). --- ### **Step 2: Express the Receiver's Velocity in Vector Components** **Given:** - Receiver's speed = 9 yards/s - Angle = 45° to the right of straight downfield (i.e., 45° from the y-axis towards x-axis) **Calculation:** - \( v_x = 9 \cos(45°) = 9 \times \frac{\sqrt{2}}{2} = 6.36 \) yards/s - \( v_y = 9 \sin(45°) = 9 \times \frac{\sqrt{2}}{2} = 6.36 \) yards/s **Vector Form:** - **Receiver's velocity** \( \vec{v}_r = (6.36\, \hat{i}, 6.36\, \hat{j}) \) yards/s **Diagram:** ``` ^ | / | / | / v_r (9 yards/s at 45°) | / | / |/ +------------------> ``` --- ### **Step 3: Find the Displacement Vector and Write the Initial Velocity Expression** **Displacement Vector (from QB to Receiver):** - \( \vec{d} = (20\, \hat{i}, 15\, \hat{j}) \) yards **To throw the ball to the receiver, the **initial velocity of the ball** must have the same direction as the receiver's velocity (since the ball must "lead" the receiver).** - **Let \( \vec{v}_{ball} \) be the initial velocity vector of the ball relative to the field.** - If you want the ball to reach the receiver, you need to solve for the time \( t \) it takes for the ball to reach the receiver's location, given both start at the same time, and the receiver is moving. - The position of the receiver as a function of time: \[ \vec{r}_{receiver}(t) = (20 + 6.36t)\, \hat{i} + (15 + 6.36t)\, \hat{j} \] - The ball must cover the distance from (0,0) to the receiver's location in time \( t \), so: \[ \vec{r}_{ball}(t) = \vec{v}_{ball} t \] Set these equal for the ball to meet the receiver: \[ \vec{v}_{ball} t = (20 + 6.36t)\, \hat{i} + (15 + 6.36t)\, \hat{j} \] \[ \vec{v}_{ball} = \left(\frac{20}{t} + 6.36\right)\, \hat{i} + \left(\frac{15}{t} + 6.36\right)\, \hat{j} \] **Final Answer:** \[ \boxed{ \vec{v}_{ball} = \left(\frac{20}{t} + 6.36\right)\, \hat{i} + \left(\frac{15}{t} + 6.36\right)\, \hat{j} \quad \text{(in yards/s)} } \] where \( t \) is the time it takes for the ball to reach the receiver. --- If you know how fast the quarterback can throw, you can solve for \( t \). Otherwise, this is the **general vector expression** for the ball's initial velocity. --- **Summary of Steps:** 1. **Diagram and Coordinates**: Plotted the positions. 2. **Receiver Velocity Components**: Calculated vector components. 3. **Ball Velocity Vector**: Derived general formula for the ball's velocity. Let me know if you'd like a calculation assuming a certain ball speed, or if you want a drawn diagram!