A ball rebounds to one quarter from which it was dropped. approximate the total vertical distance the ball after being dropped from 3ft above the ground ,until it comes to rest
first bounce --- 3 ft
2nd bounce = 2(1/4)(3) = 6(1/4)
3rd bounce = 2(1/4)(3)(1/4) = (6)(1/4)^2
4th bounce = ... = (6)(1/4)^3
....
so we have
3 + 6(1/4) + 6(1/4)^2 + 6(1/4)^3 + ... to infinity
= 3 + 6(1/4) (1 + 1/4 + (1/4)^2 + ...)
from the 2nd term on, we have a geometric series
with a = 1 , r = 1/4
sum of all terms
= 3 + (6/4) S∞
= 3 + (3/2)(1/(1-1/4)
= 3 + (3/2)(4/3)
= 5
To approximate the total vertical distance the ball travels before coming to rest, we need to consider the initial drop and the subsequent rebounds.
Given that the ball is initially dropped from a height of 3ft above the ground, let's break down the distance traveled during each rebound:
1. First drop: The ball falls from 3ft above the ground to the ground below. This distance is 3ft.
2. First rebound: The ball rebounds to one quarter of the initial drop distance. Since the initial drop was 3ft, the ball rebounds up by 3ft/4 = 0.75ft.
3. Second drop: After the first rebound, the ball falls from the rebound height (0.75ft) to the ground. This distance is again 0.75ft.
4. Second rebound: The ball rebounds to one quarter of the previous drop distance, which is one quarter of 0.75ft. Thus, the ball rebounds up by 0.75ft/4 = 0.1875ft.
5. Third drop: After the second rebound, the ball falls from the rebound height (0.1875ft) to the ground. This distance again equals 0.1875ft.
Since these drops and rebounds decrease in magnitude each time, we can approximate the total distance the ball travels before coming to rest by summing up the distances:
3ft (initial drop) + 0.75ft (first rebound) + 0.75ft (second drop) + 0.1875ft (second rebound) + 0.1875ft (third drop)
Approximately, the total vertical distance the ball travels until it comes to rest is 5.875ft.
To approximate the total vertical distance traveled by the ball until it comes to rest, we need to understand the concept of mechanical energy conservation.
The ball starts with potential energy when it is dropped from a height of 3 feet above the ground. As it falls, this potential energy is converted into kinetic energy, which is then converted back into potential energy as the ball rebounds.
Since the ball rebounds to one-quarter of the height from which it was dropped, it reaches a maximum height of 3 ft / 4 = 0.75 ft above the ground after each rebound.
To calculate the total vertical distance traveled until the ball comes to rest, we can sum up the distance traveled during each ascent and descent.
The formula to calculate the distance traveled during each ascent or descent is given by 2 * height, as the ball travels the same distance up and down.
Let's calculate the distance for one complete up-and-down cycle of the ball:
Distance = 2 * height = 2 * 0.75 ft = 1.5 ft
Since the ball comes to rest when it no longer reaches a height above the ground, we can calculate the number of cycles the ball completes until it comes to rest.
To find the number of cycles, we divide the initial height of 3 ft by the rebound height of 0.75 ft:
Number of cycles = 3 ft / 0.75 ft = 4 cycles
Now we can calculate the total vertical distance traveled until the ball comes to rest by multiplying the distance per cycle by the number of cycles:
Total distance = Distance per cycle * Number of cycles = 1.5 ft * 4 = 6 ft
Therefore, the total vertical distance traveled by the ball until it comes to rest is approximately 6 feet.