A roofing tile falls from rest off the roof of a building. An observer from across the street notices that it takes 0.42 s for the tile to pass between two windowsills that are 3.08 m apart. How far is the sill of the upper window from the roof of the building?
To find the distance between the sill of the upper window and the roof of the building, we can use the equation of motion for constant acceleration:
\[d = ut + \frac{1}{2}at^2\]
Where:
- \(d\) is the distance traveled (3.08 m)
- \(u\) is the initial velocity (0 m/s, as the tile falls from rest)
- \(t\) is the time taken (0.42 s)
- \(a\) is the acceleration (which we need to find)
In this case, the initial velocity (\(u\)) is 0 m/s and the distance traveled is 3.08 m. So, the equation simplifies to:
\[d = \frac{1}{2}at^2\]
We can rearrange the equation to solve for the acceleration:
\[a = \frac{2d}{t^2}\]
Plugging in the given values, we get:
\[a = \frac{2 \times 3.08 \, \text{m}}{(0.42 \, \text{s})^2}\]
Calculating this expression gives us the acceleration.
To find the distance between the sill of the upper window and the roof of the building, we can use the equation of motion for uniformly accelerated motion:
s = ut + (1/2)at^2
Where:
s = distance
u = initial velocity (in this case, the tile starts from rest, so u = 0)
t = time taken to travel the distance
a = acceleration (due to gravity, a = 9.8 m/s^2)
In this case, we are given the time taken, t = 0.42 s, and the distance between the two windowsills, s = 3.08 m. We need to find the distance between the sill of the upper window and the roof, which we can represent as h.
Since the tile falls freely under gravity, the vertical distance d covered by the tile is equal to the distance between the windowsills, s. Therefore, we have:
d = s = 3.08 m
Now, let's find the time taken to fall between the upper window sill and the roof, t' (where t' is the time taken for the tile to fall the distance h).
Using the equation of motion, we have:
h = (1/2)gt'^2
Now, let's solve for t'. Rearranging the equation, we get:
t' = sqrt((2h)/g)
Now, we can divide the total time taken, t, into two parts:
t = t' + (0.42 - t')
Simplifying the equation, we have:
t = t' + 0.42 - t'
t = 0.42
Since t' = sqrt((2h)/g), we can substitute this into the equation and solve for h:
0.42 = sqrt((2h)/g)
Squaring both sides of the equation, we get:
0.42^2 = (2h)/g
Simplifying further, we have:
0.1764 = 2h/9.8
Multiplying both sides of the equation by 9.8, we get:
1.7292 = 2h
Dividing both sides of the equation by 2, we find:
h = 0.8646 m
Therefore, the distance between the sill of the upper window and the roof of the building is approximately 0.8646 meters.