thanks for the help with the first problem this is the last problem I have a question on...
A superball rebounds half the height it drops. The ball is dropped from a height of 176 feet. How far off the ground is the ball when it has traveled a total of 500 feet?
You do not say if you know about geometric series so with brute force:
It goes down 176 feet and bounces
total = 176
The first bounce up is 176/2 = 88 feet
total = 264
Then it also goes down 88 feet
total =352
then up 44
total = 396
then down 44
total = 440
then up 22
total = 462
then down 22
total = 484
then up 11
total = 495 when it is 11 feet high
then down 5 to reach 500
Leaving it 6 feet high.
To solve this problem, let's break it down step by step.
First, let's find out how high the ball bounces each time it rebounds. We are given that the ball rebounds half the height it drops. Since the ball is dropped from a height of 176 feet, the height it bounces to is 176/2 = 88 feet.
Next, let's figure out how many times the ball bounces in total to cover a distance of 500 feet. We know that for each bounce, the ball travels a total distance of the drop height plus the bounce height. So, in one bounce, the ball travels a distance of 176 + 88 = 264 feet. To cover a distance of 500 feet, the ball needs to bounce 500/264 = 1.893939 (approximately 1.9) times.
Since the ball cannot bounce a fraction of a time, we need to determine how far the ball travels after the whole number of bounces. After one bounce, the ball covers a distance of 264 feet. Therefore, after one bounce, the ball has traveled a total of 264 feet. Since we only need to determine the distance from the ground when the total distance traveled is 500 feet, we need to find out how much distance is left after one bounce. The remaining distance is 500 - 264 = 236 feet.
Since the ball rebounds half the height it drops, we can conclude that after one bounce, the ball will reach a height of 88 feet. Therefore, to determine the distance from the ground when the total distance traveled is 500 feet, we add the remaining distance (236 feet) and the height of one bounce (88 feet). Thus, the ball is 236 + 88 = 324 feet off the ground when it has traveled a total of 500 feet.