a fireman 50 m away from a burning building
directs a stream of water
from a ground-level fire hose
at an angle of 30 degrees
above the horizontal. If the
speed of the stream is 40m/s , at what height will the stream of water strike the building?
t = 50/40cos30
y = 40sin30t - 1/2(9.8)t^2
To find the height at which the stream of water will strike the building, we need to break down the problem into horizontal and vertical components.
First, let's find the horizontal component of the velocity. Since the angle of elevation is 30 degrees, the horizontal component is given by:
Horizontal velocity = Velocity x cos(angle)
Horizontal velocity = 40 m/s x cos(30°)
Horizontal velocity = 40 m/s x 0.866 (rounded to three decimal places)
Horizontal velocity ≈ 34.64 m/s
Now, let's find the time it takes for the water stream to reach the building.
We can use the horizontal velocity and the distance between the fireman and the building to calculate the time. The horizontal distance is 50 meters, and the horizontal velocity is approximately 34.64 m/s.
Time = Distance / Velocity
Time = 50 m / 34.64 m/s
Time ≈ 1.443 seconds
Now, let's find the height at which the stream of water will strike the building by considering the vertical component.
Vertical displacement = (Initial vertical velocity x Time) + (0.5 x acceleration x Time^2)
Since the fire hose is at ground level, the initial vertical velocity is 0 m/s, and the acceleration due to gravity is -9.8 m/s^2 (assuming we are neglecting air resistance).
Vertical displacement = (0 x 1.443) + (0.5 x -9.8 x 1.443^2)
Vertical displacement ≈ -9.8 x 1.443^2 / 2
Vertical displacement ≈ -17.84 meters
Since we took the negative sign into account, the height at which the stream of water will strike the building is approximately 17.84 meters below the fireman's position.
Therefore, the stream of water will strike the building at a height of about 17.84 meters above ground level.