Final answer:
The total distance traveled by a dropped ball that bounces to a reduced height each time can be found by summing the distances covered in each descent and subsequent ascent. The formula requires the initial drop height and the reduction ratio after each bounce.
Step-by-step explanation:
The formula to calculate the total distance traveled by a ball dropped from a height, which then rises to a reduced height after bouncing, is determined by the sum of all the distances it travels during its motion. This includes both the descent and the ascent phases. For a ball dropped, the initial descent is from the drop height, and it travels this distance exactly once. After the first bounce, if the ball rises to a reduced height, it will do so successively each time it bounces, traveling less distance each time.
If we know by how much the height is reduced after each bounce and the initial height the ball is dropped from, we can calculate the total distance traveled by the ball as the sum of the distances covered in each phase (down and up) until the ball comes to rest or until the height reduction pattern is no longer applicable.