In physics, one learns that the time required for a ball to fall from a height of h feet (or rise to that height after a bounce) is (1/4)sqrt(h) seconds. A superball rebounds 50 % of the height from which it is dropped. Suppose that a superball is dropped from a height of 52 feet. What is the total distance that the ball has traveled, both up and down, when it strikes the ground for the 5th time?

To solve this problem, we can break down the process into several steps:

### Step 1: Understand the Problem
- A superball is dropped from a height of 52 feet.
- It rebounds 50% of the height from which it was dropped.
- We need to find the total distance traveled by the ball when it strikes the ground for the 5th time.

### Step 2: Calculate the Heights of Each Bounce
- **Initial height (h₀):** 52 feet
- **First bounce height (h₁):** \( 0.5 \times h_0 = 0.5 \times 52 = 26 \) feet
- **Second bounce height (h₂):** \( 0.5 \times h_1 = 0.5 \times 26 = 13 \) feet
- **Third bounce height (h₃):** \( 0.5 \times h_2 = 0.5 \times 13 = 6.5 \) feet
- **Fourth bounce height (h₄):** \( 0.5 \times h_3 = 0.5 \times 6.5 = 3.25 \) feet

### Step 3: Calculate the Total Distance Traveled
For each bounce, the ball travels the height of the drop and the height of the bounce. 

1. **First drop:**
   - Distance = 52 feet (down)
2. **First bounce:**
   - Distance = 26 feet (up) + 26 feet (down) = 52 feet
3. **Second bounce:**
   - Distance = 13 feet (up) + 13 feet (down) = 26 feet
4. **Third bounce:**
   - Distance = 6.5 feet (up) + 6.5 feet (down) = 13 feet
5. **Fourth bounce:**
   - Distance = 3.25 feet (up) + 3.25 feet (down) = 6.5 feet

### Step 4: Sum the Distances
Now, sum up all these distances:

- **Total distance = 52 (first drop) + 52 (first bounce) + 26 (second bounce) + 13 (third bounce) + 6.5 (fourth bounce) = 149.5 feet.**

### Conclusion
The total distance that the ball has traveled when it strikes the ground for the 5th time is **149.5 feet**.

Would you like any further details or have any other questions? Here are five related questions to consider:

1. How would the total distance change if the ball rebounded 60% instead of 50%?
2. What would be the total distance traveled if the ball was dropped from a different height, say 100 feet?
3. How does the time spent in each bounce relate to the total distance traveled?
4. What is the effect of air resistance on the total distance traveled by the ball?
5. Can this problem be solved using a geometric series? How?

**Tip:** In physics, problems involving repeated actions, such as bouncing balls, often benefit from understanding geometric sequences and series, which can simplify the calculation of total distances or times.

Learn how to calculate the total distance traveled by a superball dropped from a height of 52 feet, bouncing with 50% rebound height, until it strikes the ground for the 5th time. This problem involves kinematics and geometric sequences, suitable for high school physics students.

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