Final answer:
The ball will travel a total distance of 48 m before it comes to rest.
Step-by-step explanation:
When a ball is dropped from a height and bounces, the distance traveled by the ball can be calculated by adding up the distances covered during each bounce. In this case, the ball reaches a height that is half of the previous bounce. Let's consider the example given, where the ball is dropped from a height of 1 m and bounces back to 0.8 m, then bounces to 0.5 m, and finally to 0.2 m. The total distance traveled by the ball can be calculated as follows:
- For the first drop, the ball falls 1 m.
- For the first bounce, the ball covers a distance of (1 - 0.8) m = 0.2 m.
- For the second bounce, the ball covers a distance of (0.8 - 0.5) m = 0.3 m.
- For the third bounce, the ball covers a distance of (0.5 - 0.2) m = 0.3 m.
Therefore, the total distance traveled by the ball in this example is 1 m + 0.2 m + 0.3 m + 0.3 m = 1.8 m.
Applying the same logic to the given question, the ball is dropped from a height of 24 m. Each bounce covers half the distance of the previous bounce. Therefore, the total distance traveled by the ball can be calculated as follows:
- For the first drop, the ball falls 24 m.
- For the first bounce, the ball covers a distance of (24 - 12) m = 12 m.
- For the second bounce, the ball covers a distance of (12 - 6) m = 6 m.
- For the third bounce, the ball covers a distance of (6 - 3) m = 3 m.
- For the fourth bounce, the ball covers a distance of (3 - 1.5) m = 1.5 m.
- For the fifth bounce, the ball covers a distance of (1.5 - 0.75) m = 0.75 m.
- And so on, until the ball comes to rest.
Calculating the total distance traveled by the ball:
- Distance = 24 m + 12 m + 6 m + 3 m + 1.5 m + 0.75 m + ...
- This is a geometric series with a common ratio of 1/2.
- Applying the formula for the sum of an infinite geometric series:
Total distance = 24 m / (1 - 1/2) = 24 m / (1/2) = 24 m * 2 = 48 m.