Chap. 7, #36. (II) A ball is dropped from a height of 1.50 m and rebounds to a height of 1.20 m. Approximately how many rebounds will the ball make before losing 90% of its energy?
change that fraction to 1.2/1.5 or 4/5. I misread.
I don't understand where you got .1 and then -1 from. Can you explain that?
energy retained each bounce: 1.2/1.4=6/7
(6/7)^n=.1
log of each side
n log (6/7)= -1
solve for n.
To determine the number of rebounds the ball will make before losing 90% of its energy, we need to use the concept of the coefficient of restitution. The coefficient of restitution, denoted by the symbol "e," is a measure of how bouncy an object is. It quantifies the ratio of the final velocity to the initial velocity after a collision. In this case, the collision is between the ball and the ground.
The relationship between the initial height (h1), the rebound height (h2), and the coefficient of restitution (e) can be expressed as:
h2 = e^2 * h1
Given that the initial height (h1) is 1.50 m and the rebound height (h2) is 1.20 m, we can find the coefficient of restitution (e) as follows:
e^2 = h2 / h1
Substituting the values, we have:
e^2 = 1.20 / 1.50
Now, let's solve for e:
e = sqrt(1.20 / 1.50)
Using a calculator, we find e ≈ 0.87178
Now, we need to determine at what point the ball will have lost 90% of its energy. Since the coefficient of restitution is a measure of energy conservation, losing 90% of energy is equivalent to having only 10% of energy remaining. Therefore, we can find the number of rebounds by repeatedly multiplying the rebound height by the coefficient of restitution until it reaches or falls below 10% of the initial height.
Let's denote the initial height as h0 (1.50 m) and the remaining energy as E. At each rebound, the remaining energy is given by:
E = e^2 * E_previous
We want to find the number of rebounds for which E falls below 10% of h0. Here's how you can calculate it:
1. Initialize a variable "reboundCount" to 0.
2. Initialize a variable "E_previous" to 1 (which represents 100% of the initial energy).
3. Start a loop that continues until E falls below 10% of h0.
4. Inside the loop, calculate the new value of E using the formula: E = e^2 * E_previous.
5. Increment the "reboundCount" variable by 1.
6. Update the value of E_previous to the current value of E.
7. Repeat steps 4-6 until E falls below 10% of h0.
8. Once the loop stops, the final value of "reboundCount" will represent the number of rebounds.
You can implement this algorithm in any programming language or use a spreadsheet application like Excel to perform the calculations automatically.