A) A ball is dropped from "M" meters above a flat surface. Each time the ball hits the surface after falling a distance, h, it rebounds a distance d, where 0<r<1. Find the formula that represents the total distance the ball travels up and down.
B) Use the formula in part A to find the total distance if M=20 meters and each bounce is two-thirds of the height of the previous bounce.
I think I should be using the Geometric Series here I just cannot figure out how to write out the representing formula. Any help is appreciated.
Sn=a/1-r will be used here
where a is the first term.
i hope it is helpful!
http://www.classzone.com/eservices/home/pdf/student/LA211DBD.pdf
Thank you. Seems I was on the right track. Thank you again!
To find the formula that represents the total distance the ball travels up and down, let's break down the problem into smaller parts.
1) First, let's determine the distance the ball travels before the first bounce. The ball is dropped from a height of "M" meters, so it falls "M" meters before hitting the surface. Since the ball rebounds a distance of "d" after each bounce, the total distance the ball travels before the first bounce is "M + d".
2) After the first bounce, the ball reaches a height of two-thirds of the previous height. So, the ball will bounce to a height of (2/3)M after the first bounce. It will then fall from this height, reach the surface, and rebound a distance of "d" again. Therefore, the distance the ball travels in the second fall and rebound is ((2/3)M + d).
3) For each subsequent fall and rebound, the ball continues to reduce its height by two-thirds. So, the distance traveled in the n-th fall and rebound is ((2/3)^n * M + d).
Now that we have the distances for each fall and rebound, we can sum them together to find the total distance traveled. To sum an infinite geometric series, we can use the formula:
S = a / (1 - r),
where:
S is the total sum of the series,
a is the first term,
r is the common ratio (in this case, (2/3)).
Applying this formula to our situation, the total distance the ball travels up and down, represented by the formula S, is given by:
S = (M + d) / (1 - (2/3)).
Simplifying the formula gives:
S = 3(M + d) / (3 - 2).
Now let's move on to part B.
To find the total distance if M = 20 meters and each bounce is two-thirds of the height of the previous bounce, we can substitute these values into the formula we derived earlier:
S = 3(20 + d) / (3 - 2).
Since we do not have a specific value for "d" provided in the question, we cannot calculate the exact total distance. However, you can substitute different values of "d" to find the total distance for those specific cases.