A fire horse ejects a stream of water at an angle of 31.3 above the horizontal. The water leaves the nozzle with a speed of 28.8 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 

To solve this problem, we can model the water as a projectile. The goal is to find how far the fire hose should be located from the building to hit the highest possible fire. This involves calculating the horizontal range of the projectile at the point where the water reaches its maximum height.

### Step-by-Step Solution:

1. **Given values**:
   - Angle of projection: \( \theta = 31.3^\circ \)
   - Speed of the water: \( v_0 = 28.8 \, \text{m/s} \)
   - Acceleration due to gravity: \( g = 9.81 \, \text{m/s}^2 \)

2. **Find the vertical and horizontal components of the velocity**:
   - Vertical component: \( v_{0y} = v_0 \sin(\theta) \)
   - Horizontal component: \( v_{0x} = v_0 \cos(\theta) \)

   \[
   v_{0y} = 28.8 \times \sin(31.3^\circ)
   \]
   \[
   v_{0x} = 28.8 \times \cos(31.3^\circ)
   \]

3. **Find the time to reach the maximum height**:
   At the maximum height, the vertical velocity will be zero, so we can use the following equation:

   \[
   v_y = v_{0y} - g t
   \]

   At the maximum height, \( v_y = 0 \), so:

   \[
   0 = v_{0y} - g t_{\text{up}}
   \]

   Solving for the time to reach the maximum height \( t_{\text{up}} \):

   \[
   t_{\text{up}} = \frac{v_{0y}}{g}
   \]

4. **Find the total time of flight**:
   The total time of flight \( t_{\text{total}} \) is twice the time to reach the maximum height, because the time going up equals the time coming down:

   \[
   t_{\text{total}} = 2 \times t_{\text{up}} = 2 \times \frac{v_{0y}}{g}
   \]

5. **Find the horizontal range**:
   The horizontal range \( R \) is the distance the water travels in the horizontal direction during the entire flight. It can be calculated using:

   \[
   R = v_{0x} \times t_{\text{total}} = v_{0x} \times 2 \times \frac{v_{0y}}{g}
   \]

Now let's calculate each step.The fire hose should be located approximately **75.07 meters** away from the building to hit the highest possible point of the fire.

Would you like more details on any part of the calculation, or have any questions?

Here are 5 related questions to explore this further:
1. How high does the water reach at its maximum point?
2. What happens to the range if the angle is increased to 45 degrees?
3. How would the range change if the nozzle speed is increased to 35 m/s?
4. What is the time it takes for the water to hit the ground after reaching maximum height?
5. How does air resistance affect the range of the projectile?

**Tip**: The maximum range for a projectile (neglecting air resistance) occurs when the launch angle is 45°.

A fire hose ejects a stream of water at an angle of 31.3° above the horizontal. The water leaves the nozzle with a speed of 28.8 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?

Learn how to calculate the horizontal range of water projected from a fire hose at a 31.3° angle with a speed of 28.8 m/s. This problem covers key concepts in projectile motion, including velocity decomposition and kinematic equations. Suitable for high school students in Grades 10-12.

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