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We are given the following information:

- Acceleration \( a(t) = -32 \, \text{ft/s}^2 \) (due to gravity).
- Initial height \( h_0 = 5 \, \text{ft} \).
- Initial velocity \( v_0 = 88 \, \text{ft/s} \).

We need to determine the maximum height the ball will reach. Here's how to approach the problem:

### Step 1: Use kinematic equations
The velocity of the ball as a function of time can be expressed as:

\[
v(t) = v_0 + a \cdot t
\]

Substituting the values we know:

\[
v(t) = 88 - 32t
\]

The ball reaches its maximum height when the velocity is zero, i.e., when \( v(t) = 0 \):

\[
88 - 32t = 0
\]

Solving for \( t \):

\[
t = \frac{88}{32} = 2.75 \, \text{seconds}
\]

### Step 2: Find the position at maximum height
The position of the ball as a function of time can be given by the following equation:

\[
h(t) = h_0 + v_0 t + \frac{1}{2} a t^2
\]

Substituting the known values:

\[
h(t) = 5 + 88 \cdot 2.75 + \frac{1}{2} (-32) \cdot (2.75)^2
\]

Let's calculate the values step-by-step:

\[
h(t) = 5 + 242 + \frac{1}{2} (-32) \cdot 7.5625
\]
\[
h(t) = 5 + 242 - 121
\]
\[
h(t) = 126 \, \text{feet}
\]

So, the ball will reach a maximum height of **126 feet**.

Let me know if you'd like further details on this! Here are five related questions that you might find interesting:

1. How would the maximum height change if the initial velocity was 100 ft/s instead of 88 ft/s?
2. What would be the total time the ball spends in the air before it returns to the ground?
3. How would the inclusion of air resistance affect the maximum height?
4. What is the significance of the negative acceleration in the problem?
5. How can this same problem be solved using energy conservation principles?

**Tip:** In kinematic problems involving constant acceleration, finding when the velocity is zero helps to determine the point of maximum or minimum position (height or depth).

A ball is thrown vertically upward from a height of 5 feet with an initial velocity of 88 feet per second. How high will the ball go?

Learn how to calculate the maximum height of a ball thrown upward with an initial velocity of 88 ft/s from a height of 5 feet. Using kinematic equations, this step-by-step solution covers key concepts like acceleration and velocity. Suitable for high school students in Grades 9-12.

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