A football is punted with an initial velocity of \(27.5 \mathrm{~m} / \mathrm{s}\) and an initial angle of \(56.7^{\circ} .\) What is its hang time (the time until it hits the ground again)?

#tag_title#Step 2: Determine the vertical component of velocity#tag_content#Using the given initial velocity and angle in radians, we can find the vertical component of velocity (v_y) using the sine function: $$ v_y = v_0 \sin(\theta) $$ Where: $v_y$: vertical component of velocity $v_0$: initial velocity (40 m/s) $\theta$: angle in radians Plug in the values: $$ v_y = 40 \sin(56.7 \times \frac{\pi}{180}) $$ #tag_title#Step 3: Calculate the time to reach the highest point#tag_content#The time it takes for the football to reach its highest point can be calculated using the following equation: $$ t_{\text{up}} = \frac{v_y}{g} $$ Where: $t_{\text{up}}$: time to reach the highest point $v_y$: vertical component of velocity $g$: acceleration due to gravity (9.81 m/s²) Plug in the values: $$ t_{\text{up}} = \frac{v_y}{9.81} $$ #tag_title#Step 4: Calculate the total hang time#tag_content#The total hang time is twice the time it takes to reach the highest point since the time it takes to fall back to the ground will be the same as the time taken to reach the highest point: $$ t_{\text{hang}} = 2t_{\text{up}} $$ Plug in the value of $t_{\text{up}}$: $$ t_{\text{hang}} = 2(\frac{v_y}{9.81}) $$ Solve for $t_{\text{hang}}$ to find the total hang time of the punted football.