A falling stone takes 0.320 s to pass a window 2.30 m high. From what height (in meters) above the top of the window did the stone fall? I have tried this so many times and I cannot figure it out.
To solve this problem, we can use the equations of motion for vertical motion under constant acceleration. The equation we will use is:
y = ut + (1/2)at²
Where:
y = vertical displacement
u = initial velocity (which is zero for a falling stone)
t = time
a = acceleration due to gravity (approximately -9.8 m/s²)
In this case, the stone starts from rest, so the initial velocity u is equal to 0 m/s. We need to find the vertical displacement y.
Given:
t = 0.320 s
a = -9.8 m/s²
y = 2.30 m
Plugging in these values into the equation, we have:
2.30 = 0 + (1/2)(-9.8)(0.320)²
Simplifying the equation, we get:
2.30 = -1.568t²
Now, we can solve for t² by dividing both sides of the equation by -1.568:
t² = 2.30 / -1.568
t² ≈ -1.467
Since time cannot be negative, we discard the negative value:
t² ≈ 1.467
Now, we can find the height from which the stone fell by substituting the value of t² into the equation:
y = ut + (1/2)at²
y = 0 + (1/2)(-9.8)(1.467)
Simplifying the equation, we get:
y ≈ -6.796 m
Since height cannot be negative, we discard the negative value:
y ≈ 6.796 m
Therefore, the stone fell from a height of approximately 6.796 meters above the top of the window.
To solve this problem, we'll use kinematic equations of motion. The key equation we'll use is:
h = ut + 0.5 * gt^2
Where:
h is the total height
u is the initial velocity (which we can assume to be zero, as the stone is falling)
g is the acceleration due to gravity (approximately 9.8 m/s^2)
t is the time taken to fall
In this case, we know the total time it took for the stone to fall past the window, which is 0.320 seconds. We'll substitute this value into the equation.
First, let's solve for the height from the window to the ground (h1):
h1 = 0.5 * g * t^2
h1 = 0.5 * 9.8 * (0.320^2)
h1 = 0.5 * 9.8 * 0.1024
h1 = 4.9984 meters
Now, let's find the height from the ground to the top of the window (h2):
h2 = 2.30 - h1
h2 = 2.30 - 4.9984
h2 = -2.6984 meters
Since the height cannot be negative, it means that the stone fell from a height above the top of the window. Therefore, the stone fell from a height of 2.6984 meters above the top of the window.
Keep in mind that the negative sign just indicates the relative position below the top of the window.