A ball is thrown 12 meters in the air (so that the initial up-and-down distance is 24 meters). The ball rebounds 95% of the distance it falls. What is the total vertical distance traveled by the ball before it stops bouncing?
240 meters
To find the total vertical distance traveled by the ball before it stops bouncing, we need to consider both the distance the ball travels when it's thrown in the air and the subsequent bounces.
Let's break down the problem:
1. The ball is thrown 12 meters in the air, so the initial up-and-down distance is 24 meters.
2. The ball rebounds 95% of the distance it falls. This means that after each bounce, the ball reaches 95% of the previous height.
Now, let's calculate the total distance traveled by the ball before it stops bouncing:
For the initial throw:
- The ball travels 12 meters up.
- The ball then falls back down and travels an additional 12 meters down to the ground.
- So, the total distance traveled for the initial throw is 12 meters up + 12 meters down = 24 meters.
For the first bounce:
- The ball rebounds 95% of the distance it falls. Since it fell 12 meters, it now rebounds 0.95 * 12 meters = 11.4 meters.
- It then falls back down and travels an additional 11.4 meters down before bouncing up again.
- So, the total distance traveled for the first bounce is 11.4 meters up + 11.4 meters down = 22.8 meters.
Each subsequent bounce will have a similar pattern, where the ball rebounds 95% of the previous fall distance. The ball will continue to bounce until it reaches a point where its rebound height is negligible.
To calculate the total distance, we can set up a series with the distances of the bounces:
24 + 22.8 + 21.66 + 20.58 + ...
With a 95% rebound after each bounce, we see that each term is multiplied by 0.95 compared to the previous term.
To find the infinite sum of this series, we can use the formula for the sum of an infinite geometric series:
Sum = a / (1 - r)
where:
a = first term = 24
r = common ratio = 0.95
Plugging these values into the formula, we get:
Sum = 24 / (1 - 0.95) = 24 / 0.05 = 480 meters
Therefore, the total vertical distance traveled by the ball before it stops bouncing is 480 meters.
Where is come 0.095
Well, let's bounce right into this problem! If the initial up-and-down distance is 24 meters and the ball rebounds 95% of the distance it falls, then we can calculate the total vertical distance traveled by the ball.
First, let's figure out the distance the ball rebounds after each bounce. If the ball falls down 12 meters, it will rebound 95% of that distance, which is 11.4 meters.
Now, let's see how far the ball travels in total. From the initial throw, the ball goes up 12 meters, comes back down 12 meters, rebounds 11.4 meters, comes back down 11.4 meters, and so on. We keep adding these distances until the ball stops bouncing.
To find the total vertical distance traveled, we can use the formula for an arithmetic series:
S = (n/2)(a + l)
Where S is the total distance, n is the number of terms, a is the first term, and l is the last term.
In this case, the first term is 12 meters, the common difference is 12 - 11.4 = 0.6 meters (since the ball falls 12 meters and rebounds 11.4 meters), and we want to find the total distance, so let's call that T.
T = (n/2)(12 + l)
Now, we know that the distance of the last term, l, will be 0 since the ball stops bouncing. So we can rewrite the equation as:
T = (n/2)(12)
Now, let's find the number of terms, n. The initial up-and-down distance is 24 meters, and each bounce involves a fall and a rebound, which we can consider as one term. So, the total number of terms will be half of the up-and-down distance.
n = 24/2 = 12
Plugging n and a into our equation, we get:
T = (12/2)(12) = 6 * 12 = 72 meters
So, the total vertical distance traveled by the ball before it stops bouncing is 72 meters. And that, my friend, is how we bounce our way to the answer!
total distance
= 24 + 24(.95) + 24(.95)^2 + ....
an infinite GS, where a = 24 and r = .95
sum(all terms) = a/(1-r)
= 24/(1-.95)
= 24/.05 = 480 metres