a ball is dropped from a height of 3 meters. It bounces to 2/3 of the height from which it fell at each boune.find distance travelled
15
21
75.78
To find the total distance traveled by the ball, we need to consider two parts: the distance traveled while the ball is falling and the distance traveled while the ball is bouncing back up.
The distance traveled while the ball is falling can be calculated using the formula for distance traveled by a freely falling object:
d = (1/2) * g * t^2
Where:
d = distance traveled
g = acceleration due to gravity (approximately 9.8 m/s^2)
t = time
Since the ball is dropped from a height of 3 meters, the final distance traveled while falling will be 3 meters.
Now, let's determine the number of bounces the ball will make. Each time the ball bounces, it reaches 2/3 of its previous height. Let's represent the number of bounces as 'n'.
At the first bounce, the ball reaches a height of (2/3) * 3 = 2 meters. At the second bounce, the ball reaches a height of (2/3) * 2 = 4/3 meters. At the third bounce, the ball reaches a height of (2/3) * (4/3) = 8/9 meters. This pattern continues, and we can see that the height of each bounce is getting smaller.
To find the total distance traveled while bouncing, we can calculate the sum of the geometric series:
S(n) = a * (1 - r^n) / (1 - r)
Where:
S(n) = total distance traveled while bouncing
a = initial term in the geometric series (2 meters)
r = common ratio (2/3)
n = number of terms (number of bounces)
The formula simplifies to:
S(n) = a * (1 - r^n) / (1 - r) = 2 * (1 - (2/3)^n) / (1 - 2/3)
To determine the number of bounces, we can consider that the ball stops bouncing when it reaches a height less than or equal to 0. Since each bounce is 2/3 of the previous height, the number of bounces can be determined by solving the inequality:
(2/3)^n * 3 ≤ 0
Simplifying the inequality, we get:
(2/3)^n ≤ 0.5
To solve this inequality, we can take the logarithm of both sides:
log[(2/3)^n] ≤ log0.5
n * log(2/3) ≤ log(0.5)
n ≤ log(0.5) / log(2/3)
Using a calculator, we find that n is approximately 3.8.
Since the number of bounces must be a whole number, the ball will make a total of 3 bounces.
Now, we can calculate the total distance traveled while bouncing:
S(3) = 2 * (1 - (2/3)^3) / (1 - 2/3) = 2 * (1 - 8/27) / (1/3) = 2 * (19/27) / (1/3) = 38/9 meters.
Therefore, the total distance traveled by the ball is 3 meters (while falling) + 38/9 meters (while bouncing) = 65/9 meters, which is approximately 7.22 meters.