Solving a word problem using a quadratic function to determine the time a soccer ball remains airborne.

Problem Statement:

A soccer player kicks a ball with an initial velocity of 32 feet per second from a starting height of 0 feet. Use the vertical motion model, \( h = -16t^2 + vt + s \), where \( v \) is the initial velocity in feet/second and \( s \) is the starting height in feet, to calculate how long the ball will be in the air.

Solution:

#### Step 1: Understand the given information
- Initial velocity, \( v = 32 \) feet/second.
- Starting height, \( s = 0 \) feet.
- The vertical motion model is given by:
\[
h = -16t^2 + vt + s
\]
Substituting the given values:
\[
h = -16t^2 + 32t + 0
\]
Simplifying:
\[
h = -16t^2 + 32t
\]

#### Step 2: Determine when the ball hits the ground
The ball hits the ground when its height \( h \) is 0. Therefore, we need to solve the equation:
\[
-16t^2 + 32t = 0
\]

#### Step 3: Factor the quadratic equation
Factor out the common term \( t \):
\[
t(-16t + 32) = 0
\]
This gives us two solutions:
\[
t = 0 \quad \text{or} \quad -16t + 32 = 0
\]

#### Step 4: Solve for \( t \)
1. For \( t = 0 \):
This corresponds to the initial time when the ball is kicked (starting point).

2. For \( -16t + 32 = 0 \):
Solve for \( t \):
\[
-16t + 32 = 0
\]
\[
-16t = -32
\]
\[
t = \frac{-32}{-16}
\]
\[
t = 2
\]

#### Step 5: Interpret the results
- \( t = 0 \) is the starting time.
- \( t = 2 \) is the time when the ball hits the ground after being kicked.

#### Step 6: Calculate the total time the ball is in the air
The ball is in the air from \( t = 0 \) to \( t = 2 \). Therefore, the total time the ball is in the air is:
\[
2 - 0 = 2 \text{ seconds}
\]

Final Answer:

\[
\boxed{2}
\]