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.