The punter on a football team tries to kick a football so that it stays in the air for a long "hang time." If the ball is kicked with an initial velocity of 23.0 m/s at an angle of 58.5° above the ground, what is the "hang time"?

The hang time is twice the time it takes for the vertical velocity component (which in initially 23.0 sin 58.5 m/s) to reach zero. That equals twice the time it takes to reach its highest elevation.

T = 2 Vo sin 58.5/g

g is the acceleration of gravity.

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To calculate the hang time of the football, we need to find the time it takes for the ball to reach its highest point in the air.

Step 1: Analyze the given information.
- Initial velocity (v0) = 23.0 m/s
- Launch angle (θ) = 58.5°

Step 2: Split the initial velocity into horizontal and vertical components.
- The vertical component (v₀y) is given by v₀y = v₀ * sin(θ).
- The horizontal component (v₀x) is given by v₀x = v₀ * cos(θ).

Step 3: Calculate the time taken to reach the maximum height.
- The formula for the time taken to reach the maximum height is t = v₀y / g, where g is the acceleration due to gravity (approximately 9.8 m/s²).

Let's calculate the hang time step by step.

Step 4: Calculate the vertical component of the initial velocity.
v₀y = v₀ * sin(θ)
= 23.0 m/s * sin(58.5°)
≈ 19.97 m/s

Step 5: Calculate the time taken to reach the maximum height.
t = v₀y / g
= 19.97 m/s / 9.8 m/s²
≈ 2.04 s

So, the hang time of the football is approximately 2.04 seconds.

To calculate the hang time of a football kicked by a punter, we need to break down the motion into vertical and horizontal components.

First, let's find the vertical component of the initial velocity (Viy) using the given angle of 58.5° and the initial velocity (V0) of 23.0 m/s:

Viy = V0 * sin(θ)
Viy = 23.0 m/s * sin(58.5°)
Viy ≈ 19.3 m/s (rounded to one decimal place)

Next, let's find the time (t) it takes for the football to reach the maximum height. At the maximum height, the vertical component of velocity becomes zero.

Using the vertical motion equation:
Vfy = Viy - g * t
0 = 19.3 m/s - 9.8 m/s^2 * t

Solving for t:
t = Viy / g
t = 19.3 m/s / 9.8 m/s^2
t ≈ 2.0 seconds (rounded to one decimal place)

Since the ball reaches its maximum height in half the hang time, the total hang time (T) is:

T = 2 * t
T = 2 * 2.0 s
T = 4.0 seconds

Therefore, the hang time of the football is approximately 4.0 seconds.