A ball is dropped from the top of a building. The balls fall 3 times farther during its last second of freefall than it did during its third second of freefall. The ball fell 4.9 meters during its first second, how tall is the building and how much time is the ball in the air?
Answer d = 313.6 m
distance fallen: s(t) = 4.9t^2
during 3rd second, it fell 4.9(3^2-2^2) = 24.5m
so, during its final second, it fell 73.5m
73.5 = 4.9 (t^2 - (t-1)^2)
t = 8
so, the building height is 4.9*8^2 = 313.6
To solve this problem, we can use the equations of motion for an object in freefall. Let's break it down step by step:
Step 1: Find the distance the ball traveled during its last second of freefall.
Given that the ball falls 3 times farther during its last second of freefall than it did during its third second, we can set up the following equation:
Distance during the last second = 3 * Distance during the third second.
Step 2: Calculate the distance the ball traveled during each second of freefall.
Given that the ball fell 4.9 meters during its first second, we know that:
Distance during the first second = 4.9 meters.
Distance during the second second = 2 * Distance during the first second.
Distance during the third second = 2 * Distance during the second second.
Step 3: Calculate the total distance the ball traveled during its time in the air.
The total distance can be calculated using the following formula:
Total distance = Distance during the first second + Distance during the second second + Distance during the third second + Distance during the last second.
Step 4: Determine the time the ball is in the air.
Since the ball is dropped from rest, the time it takes to reach the ground can be calculated using the equation:
Time = √(2 * distance / acceleration due to gravity), where acceleration due to gravity is approximately 9.8 m/s².
Now, let's calculate the values:
Step 1: Distance during the last second = 3 * Distance during the third second
We do not yet know the distance during the third second, so let's calculate that first.
Step 2: Calculate the distances:
Distance during the first second = 4.9 meters
Distance during the second second = 2 * Distance during the first second = 2 * 4.9 meters = 9.8 meters
Distance during the third second = 2 * Distance during the second second = 2 * 9.8 meters = 19.6 meters
Step 1 (continued): Distance during the last second = 3 * Distance during the third second
Distance during the last second = 3 * 19.6 meters = 58.8 meters
Step 3: Calculate the total distance:
Total distance = Distance during the first second + Distance during the second second + Distance during the third second + Distance during the last second
Total distance = 4.9 meters + 9.8 meters + 19.6 meters + 58.8 meters = 93.1 meters
Step 4: Determine the time:
Time = √(2 * distance / acceleration due to gravity)
Time = √(2 * 93.1 meters / 9.8 m/s²) = √(18.98) seconds ≈ 4.36 seconds
Therefore, the height of the building is approximately 93.1 meters, and the time the ball is in the air is approximately 4.36 seconds.