A ball is thrown 16 m into the air. The ball falls, rebounds to half its previous height, and falls again. If the ball continues to rebound and fall in this manner, find the total distance the ball travels until it hits the ground for the sixth time.
16 up
16 down
ground 1
8 up
8 down
ground 2
4 up
4 down
ground 3
2 up
2 down
ground 4
1 up
1 down
ground 5
0.5 up
0.5 down
ground 6
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To find the total distance the ball travels until it hits the ground for the sixth time, we need to sum up the distances traveled during each rebound and fall.
Let's break down the problem into individual bounces:
1. The initial throw: The ball is thrown 16 m into the air.
2. The ball falls for the first time: The ball comes down from its highest point and covers a distance of 16 m.
3. The ball rebounds to half its previous height: This means it reaches a height of 8 m.
4. The ball falls for the second time: It falls from its height of 8 m, covering a distance of 8 m.
5. The ball rebounds to half its previous height: It reaches a height of 4 m.
6. The ball falls for the third time: It falls from its height of 4 m, covering a distance of 4 m.
7. The ball rebounds to half its previous height: It reaches a height of 2 m.
8. The ball falls for the fourth time: It falls from its height of 2 m, covering a distance of 2 m.
9. The ball rebounds to half its previous height: It reaches a height of 1 m.
10. The ball falls for the fifth time: It falls from its height of 1 m, covering a distance of 1 m.
11. The ball rebounds to half its previous height: It reaches a height of 0.5 m.
After five bounces, the ball has traveled a total distance of:
16 + 8 + 4 + 2 + 1 = 31 meters.
Now, to find the distance traveled until the sixth bounce, we need to consider just the fall (without the rebound) from a height of 0.5 m. The ball travels the same distance up as it did down during the last fall. So, the total distance traveled during the sixth fall is 0.5 × 2 = 1 m.
Therefore, the total distance traveled until the ball hits the ground for the sixth time is:
31 + 1 = 32 meters.