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 as it leaves the hose is 40.0 m/s at what height will the stream of water strike the building?
33.17
To find the height at which the stream of water will strike the building, we can analyze the vertical component of the water's velocity.
Given:
- Distance between the fireman and the building (horizontal distance): 50 m
- Angle above the horizontal at which the water is directed: 30 degrees
- Speed of the water stream as it leaves the hose: 40.0 m/s
First, we need to determine the vertical component of the water stream's velocity. This can be found by using the trigonometric function sine.
Vertical component of velocity (V_y) = Velocity (V) * sine(angle)
V_y = 40.0 m/s * sin(30 degrees)
Using a calculator, we find that sin(30 degrees) = 0.5.
V_y = 40.0 m/s * 0.5 = 20.0 m/s
Now we know that the vertical component of the water stream's velocity is 20.0 m/s.
Next, we can use this vertical velocity and apply the equation of motion to find the height at which the stream strikes the building.
The equation for vertical motion is:
y = y0 + V_y0t - (1/2)gt^2
Where:
- y is the final height we want to find.
- y0 is the initial height (which is 0, as the water leaves the ground-level hose).
- V_y0 is the initial vertical velocity (which is 20.0 m/s).
- g is the acceleration due to gravity (approximately 9.8 m/s²).
- t is the time taken for the water stream to reach the building.
In this case, the time taken for the water stream to travel horizontally is the same as the time taken for the water stream to reach the building vertically. So, t is the same in both horizontal and vertical directions.
The time taken can be found using the horizontal distance and horizontal velocity of the water stream:
Time (t) = Distance (d) / Horizontal Velocity (V_x)
Given that the horizontal distance (d) is 50 m, and the horizontal velocity (V_x) can be found using the cosine of the angle:
Horizontal Velocity (V_x) = Velocity (V) * cosine(angle)
V_x = 40.0 m/s * cos(30 degrees)
Again, using a calculator, we find that cos(30 degrees) = 0.866.
V_x = 40.0 m/s * 0.866 = 34.64 m/s
Now we can calculate the time taken:
t = 50 m / 34.64 m/s
t ≈ 1.445 seconds
Now we substitute the values into the vertical motion equation:
y = 0 + 20.0 m/s * 1.445 s - (1/2)(9.8 m/s²)(1.445 s)^2
Simplifying the equation, we find:
y ≈ 14.450 m
Therefore, the stream of water will strike the building at a height of approximately 14.450 meters.