A super ball bounces by 2

Question

A super ball bounces by 2

3
of its initial bounce when
dropped on a hard surface. If it is dropped from a
height of ℎ𝑐𝑚.

30. What is the total distance the ball convers
before it finally rests
A. 243−1ℎ
B. 243h
C. 3−1ℎ
D. 3h

Answer 1

Since the ball bounces by 2/3 of its previous height, the total distance covered by the ball after its n-th bounce can be written as:

h + (2/3)h + (2/3)^2h + ... + (2/3)^(n-1)h

This is a geometric series with first term h and common ratio 2/3. The sum of a geometric series is given by:

Sum = (first term)/(1 - common ratio)

Plugging in the values, we get:

Sum = h/(1 - 2/3) = 3h

So the total distance covered by the ball before it finally rests is 3h. Therefore, the answer is D. 3h.

Answer 2

To find the total distance the ball covers before it finally rests, we need to sum up the distances of each bounce.

Let's assume the initial bounce distance is 'x'.

The first bounce will cover a distance of x.
The second bounce will cover a distance of 2x (twice the initial bounce distance).
The third bounce will cover a distance of 2(2x) = 4x (twice the distance of the second bounce).

We can observe a pattern here: each bounce covers twice the distance of the previous bounce.

So, the total distance covered by the ball before it rests can be calculated as follows:

Total distance = x + 2x + 4x + ...

Now, let's express this infinite series in a different form to make it easier to calculate:

Total distance = x + 2x + 4x + ... = x(1 + 2 + 4 + ...)

Now, we have a geometric series with a common ratio of 2 and a first term of 1.

The sum of an infinite geometric series, S, can be calculated using the formula: S = a / (1 - r), where 'a' is the first term and 'r' is the common ratio.

In our case, a = 1 and r = 2.

So, the total distance covered by the ball before it rests is:

Total distance = x(1 + 2 + 4 + ...) = x / (1 - 2) = x / (-1) = -x

Therefore, the total distance covered by the ball before it finally rests is -x.

However, the options provided for the answer do not match this result. It seems like there might be an error in the given options.