Solve the problem systematically. Follow the A football is kicked with an initial velocity of \( 25 \mathrm{~m} / \mathrm{s} \) at an angle of 45 degrees with the horizontal. Determine the time of flight, the horizontal distance, and the peak of the height of the football.

To solve the problem of a football kicked with an initial velocity of \( 25 \, \mathrm{m/s} \) at an angle of \( 45^\circ \), we can break it down into several steps: ### Step 1: Extract Known Conditions - Initial velocity, \( v_0 = 25 \, \mathrm{m/s} \) - Angle of projection, \( \theta = 45^\circ \) - Acceleration due to gravity, \( g = 9.81 \, \mathrm{m/s}^2 \) ### Step 2: Calculate the Components of the Initial Velocity The initial velocity can be broken down into horizontal and vertical components using trigonometric functions: - Horizontal component: \[ v_{0x} = v_0 \cdot \cos(\theta) \] - Vertical component: \[ v_{0y} = v_0 \cdot \sin(\theta) \] ### Step 3: Calculate Time of Flight The time of flight \( T \) for a projectile launched and landing at the same height can be calculated using the formula: \[ T = \frac{2 v_{0y}}{g} \] ### Step 4: Calculate Horizontal Distance The horizontal distance \( R \) (range) can be calculated using: \[ R = v_{0x} \cdot T \] ### Step 5: Calculate Peak Height The peak height \( H \) can be calculated using: \[ H = \frac{v_{0y}^2}{2g} \] Now, let's perform the calculations step by step. #### Step 2: Calculate the Components of the Initial Velocity Using \( \theta = 45^\circ \): \[ v_{0x} = 25 \cdot \cos(45^\circ) = 25 \cdot \frac{\sqrt{2}}{2} \] \[ v_{0y} = 25 \cdot \sin(45^\circ) = 25 \cdot \frac{\sqrt{2}}{2} \] #### Step 3: Calculate Time of Flight Now, we can calculate \( T \): \[ T = \frac{2 v_{0y}}{g} = \frac{2 \cdot (25 \cdot \frac{\sqrt{2}}{2})}{9.81} \] #### Step 4: Calculate Horizontal Distance Next, we calculate \( R \): \[ R = v_{0x} \cdot T = (25 \cdot \frac{\sqrt{2}}{2}) \cdot T \] #### Step 5: Calculate Peak Height Finally, we calculate \( H \): \[ H = \frac{v_{0y}^2}{2g} = \frac{(25 \cdot \frac{\sqrt{2}}{2})^2}{2 \cdot 9.81} \] Now, let's perform these calculations. Calculate the value by following steps: - step0: Calculate: \(25\left(\frac{\sqrt{2}}{2}\right)\) - step1: Remove the parentheses: \(25\times \frac{\sqrt{2}}{2}\) - step2: Multiply: \(\frac{25\sqrt{2}}{2}\) Calculate or simplify the expression \( 2 * (25 * (\sqrt(2)/2)) / 9.81 \). Calculate the value by following steps: - step0: Calculate: \(\frac{2\left(25\left(\frac{\sqrt{2}}{2}\right)\right)}{9.81}\) - step1: Remove the parentheses: \(\frac{2\times 25\left(\frac{\sqrt{2}}{2}\right)}{9.81}\) - step2: Remove the parentheses: \(\frac{2\times 25\times \frac{\sqrt{2}}{2}}{9.81}\) - step3: Multiply the terms: \(\frac{25\sqrt{2}}{9.81}\) - step4: Convert the expressions: \(\frac{25\sqrt{2}}{\frac{981}{100}}\) - step5: Multiply by the reciprocal: \(25\sqrt{2}\times \frac{100}{981}\) - step6: Multiply: \(\frac{25\sqrt{2}\times 100}{981}\) - step7: Multiply: \(\frac{2500\sqrt{2}}{981}\) Calculate or simplify the expression \( (25 * (\sqrt(2)/2))^2 / (2 * 9.81) \). Calculate the value by following steps: - step0: Calculate: \(\frac{\left(25\left(\frac{\sqrt{2}}{2}\right)\right)^{2}}{\left(2\times 9.81\right)}\) - step1: Remove the parentheses: \(\frac{\left(25\left(\frac{\sqrt{2}}{2}\right)\right)^{2}}{2\times 9.81}\) - step2: Remove the parentheses: \(\frac{\left(25\times \frac{\sqrt{2}}{2}\right)^{2}}{2\times 9.81}\) - step3: Multiply: \(\frac{\left(\frac{25\sqrt{2}}{2}\right)^{2}}{2\times 9.81}\) - step4: Multiply the numbers: \(\frac{\left(\frac{25\sqrt{2}}{2}\right)^{2}}{19.62}\) - step5: Convert the expressions: \(\frac{\left(\frac{25\sqrt{2}}{2}\right)^{2}}{\frac{981}{50}}\) - step6: Evaluate the power: \(\frac{\frac{625}{2}}{\frac{981}{50}}\) - step7: Multiply by the reciprocal: \(\frac{625}{2}\times \frac{50}{981}\) - step8: Reduce the numbers: \(625\times \frac{25}{981}\) - step9: Multiply: \(\frac{625\times 25}{981}\) - step10: Multiply: \(\frac{15625}{981}\) ### Step 2: Calculate the Components of the Initial Velocity The horizontal and vertical components of the initial velocity are: - \( v_{0x} \approx 17.68 \, \mathrm{m/s} \) - \( v_{0y} \approx 17.68 \, \mathrm{m/s} \) ### Step 3: Calculate Time of Flight The time of flight \( T \) is calculated as: \[ T \approx 3.60 \, \mathrm{s} \] ### Step 4: Calculate Horizontal Distance The horizontal distance \( R \) can be calculated using: \[ R = v_{0x} \cdot T \approx 17.68 \cdot 3.60 \approx 63.17 \, \mathrm{m} \] ### Step 5: Calculate Peak Height The peak height \( H \) is calculated as: \[ H \approx 15.93 \, \mathrm{m} \] ### Summary of Results - **Time of Flight**: \( T \approx 3.60 \, \mathrm{s} \) - **Horizontal Distance (Range)**: \( R \approx 63.17 \, \mathrm{m} \) - **Peak Height**: \( H \approx 15.93 \, \mathrm{m} \) These calculations provide a complete analysis of the projectile motion of the football.