A fire hose ejects a stream of water at an angle of 35.8 ° above the horizontal. The water leaves the nozzle with a speed of 21.9 m/s. Assuming that the water behaves like a projectile, how far from a building should the fire hose be located to hit the highest possible fire?

You know the distance will be the distance to the hightest point of the water path.

find time to max height:
vf=vi-gt
and vf vertical is zero at the hightest ponit, so
time=21.9*sin35.8/9.8

horizonal distance=time*21.9*cos35.8

To determine how far from the building the fire hose should be located to hit the highest possible fire, we need to analyze the water stream as a projectile in projectile motion.

Projectile motion can be broken down into horizontal and vertical components. In this case, the vertical component is influenced by gravity and the horizontal component remains constant.

First, let's find the time it takes for the water stream to reach its maximum height. This can be done using the vertical component of the initial velocity and the acceleration due to gravity.

Vertical component of initial velocity (v_y) = v × sin(θ) = 21.9 m/s × sin(35.8°) = 12.7 m/s

Acceleration due to gravity (g) = 9.8 m/s² (assuming no air resistance)

Now, we can use the following equation to find the time it takes for the water stream to reach its maximum height:

v_y = v_0y + g × t

Where v_y is the vertical component of initial velocity, v_0y is the vertical component of final velocity (which is zero at the maximum height), g is the acceleration due to gravity, and t is the time.

0 = 12.7 m/s - 9.8 m/s² × t

Solving for t:

t = 12.7 m/s / 9.8 m/s² ≈ 1.30 s

This means it takes approximately 1.30 seconds for the water stream to reach its maximum height.

Now, let's determine the vertical distance traveled by the water stream during this time. This can be calculated using the following equation:

Δy = v_0y × t + (1/2) × g × t²

Where Δy is the vertical distance, v_0y is the vertical component of initial velocity, g is the acceleration due to gravity, and t is the time.

Δy = 12.7 m/s × 1.30 s + (1/2) × 9.8 m/s² × (1.30 s)² ≈ 21.2 m

The water stream reaches a maximum height of approximately 21.2 meters.

Now, let's determine the horizontal distance traveled by the water stream. This can be calculated using the horizontal component of the initial velocity and the time it takes for the water stream to reach its maximum height.

Horizontal component of initial velocity (v_x) = v × cos(θ) = 21.9 m/s × cos(35.8°) = 17.7 m/s

Horizontal distance (d) = v_x × t

d = 17.7 m/s × 1.30 s ≈ 22.9 m

Therefore, the fire hose should be located approximately 22.9 meters away from the building to hit the highest possible fire.