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 24.2 m/s at an angle of 58.5° above the ground, what is the "hang time"?

Double the time it takes to reach zero vertical velocity component. Half the time is spenty rising to that height; the other half is spent coming back down.

To find the hang time of the football, we can use the basic principles of projectile motion. The hang time refers to the total amount of time the football remains in the air before landing back on the ground.

The hang time can be determined by calculating the time it takes for the football to reach its maximum height and then double that time to account for the time it takes for the football to descend and return to the ground.

First, we need to break the initial velocity into its horizontal and vertical components. The horizontal component remains constant throughout the flight because there is no acceleration in the horizontal direction. The vertical component is affected by gravity and can be calculated using the following equation:

Vf = Vi + at

Where:
Vf = final vertical velocity (in this case, 0 m/s at the highest point)
Vi = initial vertical velocity
a = acceleration due to gravity (approximately 9.8 m/s²)
t = time

We can determine the initial vertical velocity (Vi) by multiplying the initial velocity (24.2 m/s) by the sine of the launch angle (58.5°):

Vi = 24.2 m/s * sin(58.5°)

Next, we can use this value to find the time it takes for the football to reach its maximum height. At the highest point of its trajectory, the vertical velocity becomes zero:

Vf = Vi + at

0 = Vi + (-9.8 m/s²)t

Solving for t:

t = -Vi / (-9.8 m/s²)

Now, we can double this time to account for the descent of the football:

hang time = 2 * t

Finally, we substitute the calculated values and solve for hang time to get the answer.