In Physics class, a student is bouncing a ball and measuring the maximum height of the ball with each bounce. The ball

was originally dropped from 6 feet above the ground. The student has collected the following data so far.

Heights of a Bouncing Ball

b

h

(Bounce number) (Maximum Height)

0

6

1

4.66

2

3.63

3

2.82

4

2.19  Give a formula for a mathematical model that best fits this data. Let b be the number of the bounce, and h

be the height of that bounce.

The data provided suggests that the height of the ball decreases after each bounce, following a diminishing trend. This kind of pattern typically fits an **exponential decay model** of the form:

\[
h(b) = h_0 \cdot r^b
\]

Where:
- \( h(b) \) is the height after \( b \) bounces,
- \( h_0 \) is the initial height (in this case, 6 feet),
- \( r \) is the ratio by which the height decreases after each bounce,
- \( b \) is the bounce number.

### Step-by-Step Derivation:

1. The initial height is given as 6 feet, so \( h_0 = 6 \).
   
2. We need to determine the decay ratio \( r \). This can be done using the formula for two consecutive heights:

   \[
   r = \frac{h(b+1)}{h(b)}
   \]

   Using bounce 0 and bounce 1:
   \[
   r = \frac{4.66}{6} \approx 0.7767
   \]

3. To verify the model fits, we can check if this ratio holds for subsequent bounces:
   - Bounce 1 to 2: \( \frac{3.63}{4.66} \approx 0.778 \)
   - Bounce 2 to 3: \( \frac{2.82}{3.63} \approx 0.777 \)
   - Bounce 3 to 4: \( \frac{2.19}{2.82} \approx 0.777 \)

   The ratio \( r \) stays consistently around \( 0.777 \).

Thus, the mathematical model for the height of the ball after \( b \) bounces is:

\[
h(b) = 6 \cdot (0.777)^b
\]

### Final Formula:
\[
h(b) = 6 \cdot 0.777^b
\]

This equation models the height of the ball after \( b \) bounces.

Would you like more details or have any questions?

---

Here are 5 related questions you might consider:

1. How would the model change if the ball were dropped from a different height?
2. What happens to the ball's height as the number of bounces increases indefinitely?
3. How can we use this model to predict the height after the 10th bounce?
4. What is the physical meaning of the ratio \( r \) in this context?
5. Can this model be used to estimate the total distance the ball travels?

**Tip:** Exponential decay models are common in real-life scenarios, including physics (e.g., damping, cooling) and economics (e.g., depreciation).

In Physics class, a student is bouncing a ball and measuring the maximum height of the ball with each bounce. The ball was originally dropped from 6 feet above the ground. The student has collected the following data so far:
Heights of a Bouncing Ball
Bounce number (b): 0, 1, 2, 3, 4
Maximum Height (h): 6, 4.66, 3.63, 2.82, 2.19
Give a formula for a mathematical model that best fits this data. Let b be the number of the bounce, and h be the height of that bounce.

This problem demonstrates how to create an exponential decay model to predict the height of a bouncing ball. The ball was initially dropped from 6 feet, and the data shows decreasing heights after each bounce. The formula h(b) = 6 * 0.777^b models this behavior, ideal for high school students in Grades 9-11.

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