Suppose that a ball originally fell from a height of 2.0m and on each bounce the ball rises to 70% of the height from which it fell. Express the height reached, h , as a function of the number of bounces, n.
after bounce 1: height = .7(2)
after bounce 2: height = .7 (.7)(2) = 2 (.7)^2
after bounce 3: height = 2 (.7)^3
...
after bounce n: height = .....
How to formalize it into a equation using h as height, n as bounces
geeessh, what did I use for "height", what did I use for the number of bounces?
h=2(0.70)^n
correct
To express the height reached, h, as a function of the number of bounces, n, we need to understand the pattern of how the ball bounces and the relationship between the height and the number of bounces.
Let's break down the problem step by step:
1. The ball initially falls from a height of 2.0m. So, we can say h(0) = 2.0m, where h(0) represents the initial height.
2. On each bounce, the ball rises to 70% of the height from which it fell. This means that the height of the ball after one bounce, h(1), is 70% of the initial height, h(0). So, we can write h(1) = 0.70 * h(0).
3. For subsequent bounces, the same pattern repeats. The height after the second bounce, h(2), is 70% of the height after the first bounce, h(1). So, we can write h(2) = 0.70 * h(1).
4. Continuing this pattern, we can express the height after the nth bounce, h(n), as 70% of the height after the (n-1)th bounce, h(n-1). Thus, h(n) = 0.70 * h(n-1).
Now, we have a recursive formula for the height after each bounce, which can be used to find the height reached after any number of bounces.
To explicitly express the height as a function of the number of bounces, we can use the recursive formula to find a general expression for h(n) in terms of h(n-1), h(n-2), and so on.
Using the given information that the initial height is 2.0m, we have:
h(0) = 2.0m
h(1) = 0.70 * h(0) = 0.70 * 2.0m = 1.4m
We can apply the recursive formula to find the next few bounces:
h(2) = 0.70 * h(1) = 0.70 * 1.4m = 0.98m
h(3) = 0.70 * h(2) = 0.70 * 0.98m = 0.686m
h(4) = 0.70 * h(3) = 0.70 * 0.686m = 0.4802m
From this pattern, we can see that the height reached after each bounce decreases. The height after each bounce can be expressed as a function of n using the recursive formula:
h(n) = 0.70 * h(n-1)
Thus, the height reached after n bounces can be calculated by substituting the height after (n-1) bounces into the recursive formula and evaluating it iteratively.
Please note that this is a recursive formula and not an explicit formula. If you need an explicit formula, you would need to find a closed-form expression for h(n) using techniques like geometric progression formulas.