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?

To find the height at which the stream of water will strike the building, we need to separate the horizontal and vertical components of the stream's velocity.

Given:
Initial velocity of the stream, v_0 = 40.0 m/s
Angle of the stream above the horizontal, θ = 30 degrees

First, we'll find the horizontal component of the velocity, v_x, and the vertical component of the velocity, v_y.

v_x = v_0 * cos(θ)
v_y = v_0 * sin(θ)

Now, we can determine the time it takes for the water stream to reach the building.

The horizontal distance between the fireman and the building, d_hor = 50 m

d_hor = v_x * t
t = d_hor / v_x

Next, we can calculate the time it takes for the water stream to fall back down to the ground.

The acceleration due to gravity, g = 9.8 m/s²
The vertical distance the water stream traveled, d_ver, is what we're trying to find.

Using the equation of motion:
d_ver = (1/2) * g * t²

Now, substitute the value of t from the earlier calculation:
d_ver = (1/2) * g * (d_hor / v_x)²

Finally, we can determine the height at which the stream of water will strike the building by summing the vertical distance from the ground and the height of the fireman.

Height = d_ver + height of the fireman

Substitute the values and calculate the height.