A player kicks a football from ground level with a velocity of magnitude 31.0 m/s at an angle of 30.0 degrees above the horizontal. What is its maximum height?

The vertical component of velocity is: Vy = 31 sin 30 = 15.5 m/s

That is what determines the maximum height, H.

The maximum height is given by
sqrt(2 g H) = Vy
H = (Vy)^2/(2g)

12.0m

To find the maximum height of the football, we can use the basic principles of projectile motion. The key idea is that at the highest point of the trajectory, the vertical component of the velocity will be zero.

1. Split the velocity into its horizontal and vertical components:
The horizontal component (Vx) can be found using the formula Vx = V * cos(θ), where V is the magnitude of the velocity (31.0 m/s) and θ is the angle above the horizontal (30.0 degrees).
The vertical component (Vy) can be found using the formula Vy = V * sin(θ).

2. Determine the time it takes for the ball to reach maximum height:
At the highest point, the vertical displacement (change in height) is zero.
The formula for vertical displacement is given by: Δy = Vyo * t - 0.5 * g * t^2, where Vyo is the initial vertical velocity (Vy), t is the time, and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Since the vertical displacement is zero, we can set the equation to zero and solve for t.

0 = Vyo * t - 0.5 * g * t^2

Rearranging the equation:
0.5 * g * t^2 = Vyo * t

Dividing both sides by t:
0.5 * g * t = Vyo

Solving for t:
t = Vyo / (0.5 * g)

3. Calculate the maximum height:
Once we have the time it takes for the ball to reach maximum height, we can substitute it back into the equation for vertical displacement.
Δy = Vyo * t - 0.5 * g * t^2

Plug in the values: Vyo = Vy, g = 9.8 m/s^2, and t from step 2.

This will give you the maximum height above the ground that the football reaches.

Let's calculate the maximum height:

Step 1:
Vx = 31.0 m/s * cos(30.0 degrees)
Vy = 31.0 m/s * sin(30.0 degrees)

Step 2:
0 = Vy * t - 0.5 * g * t^2
Rearranging the equation: 0.5 * g * t = Vy
Solving for t: t = Vy / (0.5 * g)

Step 3:
Δy = Vy * t - 0.5 * g * t^2

Now, we can plug in the values and calculate the maximum height.