A fireman 46.0 m away from a burning building directs a stream of water from a ground-level fire hose at an angle of 30.0° above the horizontal. If the speed of the stream as it leaves the hose is 40.0 m/s, at what height will the stream of water strike the building?
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To find the height at which the stream of water will strike the building, we can split the initial velocity of the water into horizontal and vertical components.
Step 1: Find the vertical component of the velocity.
The vertical component can be found using the formula:
Vertical component = (initial velocity) * sin(angle)
Vertical component = (40.0 m/s) * sin(30.0°)
Vertical component = 20.0 m/s
Step 2: Find the time it takes for the water to reach the building.
We can use the equation of motion:
Distance = (initial velocity) * time - (1/2) * g * time²
Since we are interested in the vertical distance, we will use the vertical component and the acceleration due to gravity (g = 9.8 m/s²).
Distance = (20.0 m/s) * time - (1/2) * (9.8 m/s²) * time²
Rearranging the equation, we get:
(1/2) * (9.8 m/s²) * time² - (20.0 m/s) * time = 0
Using the quadratic formula, we get:
time = (-b ± √(b² - 4ac)) / (2a)
where a = (1/2) * (9.8 m/s²), b = -20.0 m/s, and c = 0.
time = (-(-20.0) ± √((-20.0)² - 4 * (1/2) * (9.8) * 0)) / (2 * (1/2) * (9.8))
time = (20.0 ± √(400 + 0)) / (9.8)
time = (20.0 ± √400) / 9.8
time = (20.0 ± 20.0) / 9.8
Taking the positive value:
time = (20.0 + 20.0) / 9.8
time = 40.0 / 9.8
time ≈ 4.08 seconds
Step 3: Find the height.
The height can be found using the formula:
Height = (vertical component) * time
Height = 20.0 m/s * 4.08 s
Height ≈ 81.6 meters
Therefore, the stream of water will strike the building at a height of approximately 81.6 meters.
To solve this problem, we can break it down into two components: horizontal and vertical.
First, let's find the horizontal distance traveled by the water stream. We can use the formula:
Horizontal distance = velocity * time
In this case, the velocity is the horizontal component of the water stream's initial velocity, which is given by the equation:
Vx = velocity * cos(θ)
Where θ is the angle of the water stream, which is 30.0° in this case. Therefore, we have:
Vx = 40.0 m/s * cos(30.0°)
Vx = 40.0 m/s * 0.866
Vx ≈ 34.64 m/s
Now, we can calculate the time it takes for the water stream to reach the building. The time can be found using the formula:
Time = Horizontal distance / velocity
In this case, the horizontal distance is the distance from the fireman to the building, which is 46.0 m. Therefore, we have:
Time = 46.0 m / 34.64 m/s
Time ≈ 1.33 s
Now, let's find the vertical distance traveled by the water stream. We can use the equation:
Vertical distance = velocity * sin(θ) * time
In this case, the vertical distance is the height at which the water stream strikes the building. Therefore, we have:
Vertical distance = 40.0 m/s * sin(30.0°) * 1.33 s
Vertical distance ≈ 21.78 m
Therefore, the water stream will strike the building at a height of approximately 21.78 meters.