A ball has bounce coefficient 0 < r < 1 if when it is dropped from height h, it bounces back to a height of rh. Suppose that such a ball is dropped from an initial height a and subsequently bounces infinitely may times. Find the total up-and-down distance in all its bouncing.
h[1 + 2r + 2r^2 + 2r^3 + ...]
= 2h(1 +r +r^2 + r^3 + ..) -h
= 2h [1/(1-r)] -h
distance= h+2hr+2hr^2+2hr^3...
distance= 2h+2hr+ 2hr^2+... -h
Isn't that an geometric series?
http://en.wikipedia.org/wiki/Geometric_progression
sum= a/(1-r) -h= 2h/(1-r) -h
check my thinking
To find the total up-and-down distance in all the bounces of the ball, we will analyze the pattern of the bounces and calculate the distance traveled in each bounce.
Given that the ball is dropped from an initial height "a", it bounces back to a height of "r*a" with each bounce, where 0 < r < 1 is the bounce coefficient. This means that after the first bounce, the ball reaches a height of r*a, then after the second bounce, it reaches a height of r*(r*a), and so on.
Mathematically, the height of the ball after n bounces can be expressed as h(n) = r^n * a.
Now, let's analyze the pattern:
- On the way down: The ball falls from height "a" to height "0" in the first bounce.
Then, it falls from height "r*a" to height "0" in the second bounce.
And so on...
- On the way up: The ball rebounds from height "0" to height "r*a" after the first bounce.
Then, it rebounds from height "0" to height "r*(r*a)" after the second bounce.
And so on...
Notice that the distance covered in each bounce (both on the way down and on the way up) is equal to the height difference between the starting and ending points.
- On the way down: The distance covered = a - 0 = a.
- On the way up: The distance covered = r * a - 0 = r * a.
Now, let's calculate the total up-and-down distance in all the bounces by summing the distances covered in each bounce:
Total distance = (a) + (r*a) + (r^2 * a) + (r^3 * a) + ...
This is an infinite geometric series with first term "a" and common ratio "r". The sum of an infinite geometric series is given by:
Sum = a / (1 - r)
Therefore, the total up-and-down distance in all the bounces is:
Total distance = a / (1 - r)
This formula gives us the exact value of the total distance traveled by the ball in terms of the initial height "a" and the bounce coefficient "r".