A fire hose ejects a stream of water at an angle of 30.9° above the horizontal. The water leaves the nozzle with a speed of 26.5 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?

Vi = 26.5 sin 30.9 = 13.6

v = Vi - g t
at top v = 0
9.81 t = 13.6 at top
t = 1.39 seconds up

u = 26.5 cos 30.9
u = 22.7

d = u t = 22.7*1.39 = 31.6 meters

To find the distance from the building, we can use the concepts of projectile motion. First, let's break down the given information:

Angle of projection: θ = 30.9°
Initial velocity: v = 26.5 m/s

In projectile motion, we can analyze the horizontal and vertical components of the motion separately.

Horizontal Motion:
The horizontal component of the velocity (Vx) remains constant throughout the motion since there is no acceleration in the horizontal direction.
Vx = velocity * cos(angle)
Vx = 26.5 m/s * cos(30.9°)
Vx = 26.5 m/s * 0.866
Vx = 22.95 m/s

Vertical Motion:
The vertical component of the velocity (Vy) changes due to the effect of gravity.
Vy = velocity * sin(angle)
Vy = 26.5 m/s * sin(30.9°)
Vy = 26.5 m/s * 0.5
Vy = 13.25 m/s

To calculate the time taken for the water to reach the highest point of its trajectory, we need to determine the time when the vertical component becomes zero.
Using the equation for vertical displacement (Δy) in projectile motion:

Δy = Vy * t + (0.5) * a * t^2

Since the vertical displacement at the highest point is zero, we have:

0 = 13.25 m/s * t + (0.5) * (-9.8 m/s^2) * t^2

Simplifying the equation:

4.9 t^2 - 13.25 t = 0

Solving this quadratic equation, we get two possible solutions: t = 0 s or t = 2.693 s.

Since time cannot be zero in this context, the water takes approximately 2.693 seconds to reach the highest point.

With the time of flight (t), we can now calculate the horizontal displacement (range) using the formula:

Range (R) = Vx * t
R = 22.95 m/s * 2.693 s
R ≈ 61.86 meters

Therefore, the fire hose should be located approximately 61.86 meters from the building to hit the highest possible fire.