A stone, initially at rest, is dropped from the top of a building. On its way down, it passes a window that is 2.52m tall. It takes 0.134 seconds to pass the window. From how far above the top of the window was the stone dropped?
Let H be the distance from the top of the building to the top of the window.
Let T be the time for the stone to reach that point. g = 9.91 m/s^2
Here is what you knmow:
H = (1/2) g T^2
H + 2.25 = (1/2) g (T + 0.134)^2
Combining the two equations, you can eliminate H
2.25 = (1/2) g [0.268 T + 0.018]
Solve that for T; then compute H with one of the original equations.
To solve this problem, we can use the equation for the motion of an object in free fall:
y = (1/2)gt^2
Where:
y is the vertical distance traveled by the object
g is the acceleration due to gravity (approximately 9.8 m/s^2)
t is the time taken to travel the distance
Given that the window is 2.52m tall and it takes 0.134 seconds for the stone to pass the window, we can plug these values into the equation to find the distance from above the top of the window the stone was dropped.
Let's calculate it step by step:
1. Determine the distance traveled by the stone when passing the window:
Using the equation, y = (1/2)gt^2, we substitute the known values:
2.52m = (1/2)(9.8 m/s^2)(0.134s)^2
Simplifying the equation:
2.52 = (1/2)(9.8)(0.017956)
2.52 = 0.5(9.8)(0.017956)
2.52 = 0.08828432
2. Find the total distance the stone traveled from above the top of the window:
Let x represent the distance from above the top of the window the stone was dropped. The total distance traveled by the stone is the sum of the distance covered before passing the window (x) and the distance covered after passing window (2.52 m).
Total distance = x + 2.52 m
3. Set up an equation using the total distance traveled and the time taken:
The total distance can be calculated using the equation d = v*t, where v is the average speed of the stone. In this case, since the stone is in free fall, the speed can be calculated using the equation v = g*t, where g is the acceleration due to gravity.
Thus, the total distance traveled can be expressed as the average speed multiplied by the time taken:
total distance = average speed * time
x + 2.52 = (0 m/s + 9.8 m/s^2 * 0.134 s
Simplifying the equation:
x + 2.52 = (9.8)(0.134)
x + 2.52 = 1.3152
4. Solve for x:
Subtract 2.52 from both sides of the equation:
x = 1.3152 - 2.52
x = -1.2048
Since the distance above the top of the window can't be negative, we conclude that the stone was dropped from 1.2048 meters above the top of the window.
Therefore, the stone was dropped from a height of approximately 1.2048 meters above the top of the window.