A rubber ball is dropped from a height of 9m and each time it strikes the ground, pounces to a height 2/3 of that from which it alls. Find the total distance travelled by the all before it comes to rest.
To find the total distance travelled by the ball before it comes to rest, we need to sum up the distances it travels during each bounce.
The ball is dropped from a height of 9m, so it travels a distance of 9m during the first fall.
After the first bounce, the ball reaches a height of (2/3) * 9m = 6m.
Then, it falls again and reaches a height of 6m during the second fall.
After the second bounce, the ball reaches a height of (2/3) * 6m = 4m.
Then, it falls and reaches a height of 4m during the third fall.
After the third bounce, the ball reaches a height of (2/3) * 4m = 8/3 m.
Then, it falls and reaches a height of 8/3 m during the fourth fall.
For each subsequent bounce, the height reached decreases by a factor of 2/3.
Let's calculate the distances travelled during the bounces and the falls:
1st fall: 9m
1st bounce: 9m
2nd fall: 6m
2nd bounce: 6m
3rd fall: 4m
3rd bounce: 4m
4th fall: 8/3 m
4th bounce: 8/3 m
5th fall: (2/3) * (8/3) m = 16/9 m
The total distance travelled by the ball is:
9m + 9m + 6m + 6m + 4m + 4m + 8/3 m + 8/3 m + 16/9 m
We can simplify this expression:
9m + 9m + 6m + 6m + 4m + 4m + 8/3 m + 8/3 m + 16/9 m = 36m + 16/3 m + 16/9 m
Now, we need a common denominator to add these fractions:
36m + 16/3 m + 16/9 m = (108/9)m + (48/9)m + 16/9 m = (108 + 48 + 16)/9 m = 172/9 m
The total distance travelled by the ball before it comes to rest is 172/9 m.