Let's break down the problem step by step: ## Given: - **Receiver's position:** 16 yards down the field (**x**), 30 yards to the quarterback's left (**y**) - **Initial velocity:** \( 41 \) miles per hour - **Launch angle:** \( 45^\circ \) upward - **Coordinates system:** - Positive \( x \): down the field - Positive \( y \): quarterback's left - Positive \( z \): upward ## Step 1: Convert Velocity to ft/s First, convert \( 41 \) miles/hour to **feet/second** for consistency with yards. \[ 41 \text{ mi/hr} \times \frac{5280 \text{ ft}}{1 \text{ mi}} \times \frac{1 \text{ hr}}{3600 \text{ s}} = 60.13 \text{ ft/s} \] ## Step 2: Find Direction in the XY Plane The receiver is: - \( x = 16 \) yards \( = 48 \) feet - \( y = 30 \) yards \( = 90 \) feet The direction in the horizontal plane (XY plane) from the quarterback to the receiver is: \[ \vec{r}_{xy} = 48\hat{\imath} + 90\hat{\jmath} \] The magnitude is: \[ |\vec{r}_{xy}| = \sqrt{48^2 + 90^2} = \sqrt{2304 + 8100} = \sqrt{10404} \approx 102 \text{ ft} \] The horizontal direction **unit vector** is: \[ \hat{u}_{xy} = \frac{48}{102}\hat{\imath} + \frac{90}{102}\hat{\jmath} = 0.4706\hat{\imath} + 0.8824\hat{\jmath} \] ## Step 3: Split Initial Velocity The ball is thrown at a **45° angle above** the horizontal. - **Horizontal component:** \( v_0\cos(45^\circ) \) - **Vertical component:** \( v_0\sin(45^\circ) \) Since \(\cos(45^\circ) = \sin(45^\circ) = \frac{\sqrt{2}}{2} \approx 0.7071\): \[ v_{xy} = 60.13 \times 0.7071 = 42.5 \text{ ft/s} \] \[ v_z = 60.13 \times 0.7071 = 42.5 \text{ ft/s} \] The horizontal velocity vector points in the direction of \(\hat{u}_{xy}\): \[ \vec{v}_{xy} = 42.5 \cdot (0.4706\hat{\imath} + 0.8824\hat{\jmath}) = 20.0\hat{\imath} + 37.5\hat{\jmath} \] ## Step 4: Write Full Initial Velocity Vector Combine horizontal and vertical components: \[ \boxed{ \vec{v}_0 = 20.0\hat{\imath} + 37.5\hat{\jmath} + 42.5\hat{k} \quad \text{(in ft/s)} } \] - \( \hat{\imath} \): down the field (x) - \( \hat{\jmath} \): to quarterback's left (y) - \( \hat{k} \): upward (z) --- **Summary in standard unit vectors:** \[ \boxed{ \vec{v}_0 = 20.0\,\hat{\imath} + 37.5\,\hat{\jmath} + 42.5\,\hat{k} \quad \text{ft/s} } \] Let me know if you want the answer in SI units (m/s) or with more significant digits!