A ball is dropped from ten feet and caught after it touches the ground on the 30 th bounce, what is the distance traveled by the ball?
To find the distance traveled by the ball, we need to calculate the sum of the distances covered during each bounce.
In this scenario, the ball is dropped from a height of ten feet, which means the first bounce covers a distance of ten feet. The ball then bounces up and comes back down for a total distance covered of 10 + 10 = 20 feet.
For the subsequent bounces, we need to consider that the ball loses some energy with each bounce due to friction and air resistance. Assuming a perfectly elastic bounce with no energy loss, each bounce covers twice the height of the previous bounce.
Starting from the second bounce, the distances covered by the ball in each bounce can be calculated as follows:
2nd bounce: 10 feet * 2 = 20 feet
3rd bounce: 20 feet * 2 = 40 feet
4th bounce: 40 feet * 2 = 80 feet
...
nth bounce: 20 * 2^(n-2) feet
Since we want to find the distance traveled on the 30th bounce, we can substitute n=30 into the equation:
30th bounce: 20 * 2^(30-2) feet = 20 * 2^28 feet
Calculating the value, we find that the distance covered on the 30th bounce is approximately 515,396,075 feet.
Now, to find the total distance traveled by the ball, we need to sum up the distances covered during each bounce. This can be done by summing a geometric series:
Total distance = 20 + 40 + 80 + ... + 20 * 2^28
To simplify the process, we can use the formula for the sum of a geometric series:
Sum = a * (1 - r^n) / (1 - r)
Where:
a = first term = 20
r = common ratio = 2
n = number of terms = 29 (since we are excluding the first term in the sum)
Plugging in the values, we can calculate the total distance traveled by the ball:
Total distance = 20 * (1 - 2^29) / (1 - 2)
Total distance ≈ 20 * (1 - 536,870,912) / (1 - 2)
Total distance ≈ 20 * (-536,870,911) / -1
Total distance ≈ 10,737,418,220 feet
Therefore, the ball travels approximately 10,737,418,220 feet in total.