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Let's break down the problem and solve it using geometric series principles.

### Part a: Height after two bounces

The ball is initially dropped from a height of \( 9 \) meters, and after each bounce, it rises to \( \frac{2}{3} \) of its previous height.

The first bounce brings the ball to:
\[
H_1 = 9 \times \frac{2}{3} = 6 \text{ meters}
\]

The second bounce brings the ball to:
\[
H_2 = 6 \times \frac{2}{3} = 4 \text{ meters}
\]

Thus, the height after two bounces is **4 meters**.

### Part b: Total distance traveled after 10 bounces

The total distance is composed of both the ball's descent and ascent. For the first drop (before the first bounce), the ball travels 9 meters downward. For each subsequent bounce, the ball travels down and then back up. 

To find the total distance traveled, we need to account for:
- The initial drop of 9 meters.
- The distances traveled up and down after each bounce.

The distance traveled after the first bounce is:
\[
\text{Up + Down} = 6 \, (\text{up}) + 6 \, (\text{down}) = 12 \, \text{meters}
\]
The distance traveled after the second bounce is:
\[
\text{Up + Down} = 4 \, (\text{up}) + 4 \, (\text{down}) = 8 \, \text{meters}
\]
This forms a geometric series where the first term is \(12\) and the common ratio is \( \frac{2}{3} \).

The sum of a geometric series is given by:
\[
S_n = \frac{a(1 - r^n)}{1 - r}
\]
where:
- \( a = 12 \) (the distance of the first bounce),
- \( r = \frac{2}{3} \) (the ratio of consecutive distances),
- \( n = 9 \) (since after the 10th bounce, the ball will have bounced 9 times).

Let's calculate the total distance after 10 bounces, including the initial drop of 9 meters.

I'll calculate this for you now.The total distance traveled by the ball when it hits the ground for the 10th time is approximately **44.06 meters**.

### Summary:
- **a)** The height after two bounces is **4 meters**.
- **b)** The total distance traveled after 10 bounces is approximately **44.06 meters**.

Would you like further clarification on any part of the solution or additional details?

Here are 5 related questions to explore:
1. How would the total distance change if the bounce ratio was different?
2. What would the height be after 5 bounces?
3. How many bounces would it take for the ball to rise less than 1 meter?
4. How would the problem change if air resistance were considered?
5. Can you derive the total distance formula if the initial height or ratio is changed?

**Tip:** Geometric series are powerful for analyzing repeated patterns, like bouncing balls or even compound interest.

A ball is dropped from a height of 9 meters and it bounces up to 2/3 of its previous height after each bounce. a) Calculate the height it will rise after two bounces. b) Calculate the total distance travelled by the ball when it hits the ground for the 10th time.

Solve a physics and math problem involving a ball bouncing to 2/3 of its previous height. Learn how to calculate the height after two bounces and the total distance traveled after 10 bounces using geometric series. Suitable for Grades 9-11 students.

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