A building’s 10th floor (34.5 m high) is blazing with fire. A fire truck arrived at the scene and the fire
men shoots water from their hose. The water leaves the hose at the speed of 29 m/s, at an angle of
63° and is held at 0.90 m from the ground. Will the water reach the fire? If so, how far from the
building should the hose be so the fire could be put out?
the equation is
y = -4.9/(29 cos63°)^2 x^2 + tan63° x + 0.9
so see where y=34.5 (if ever)
Well, let's do some calculations and clown around with the numbers! The first thing we need to find out is the time it takes for the water to reach the top of the building. We can use the information given to figure that out.
The vertical component of the water's velocity is given by V_y = V * sin(theta), where V is the initial velocity of the water and theta is the shooting angle.
So, V_y = 29 m/s * sin(63°) = 25.84 m/s.
Now, we can use this vertical component to calculate the time it takes for the water to reach the top of the building. The height of the building is 34.5 m, so we can use the equation:
34.5 m = V_y * t - 0.5 * g * t^2,
where g is the acceleration due to gravity (9.8 m/s^2).
Rearranging the equation, we get:
0.5 * g * t^2 - V_y * t + 34.5 m = 0.
Solving this quadratic equation, we find that t ≈ 2.40 s.
So, it takes approximately 2.40 seconds for the water to reach the top of the building.
Now, let's find out how far from the building the hose should be to put out the fire.
In 2.40 seconds, the water will travel horizontally a distance of:
d = V * cos(theta) * t.
Substituting the values, we get:
d = 29 m/s * cos(63°) * 2.40 s ≈ 25.32 m.
So, the hose should be approximately 25.32 meters away from the building in order for the water to reach the fire.
Now, remember that these calculations are based on idealized conditions and neglect factors like air resistance. So, it's always good to keep your clown shoes on and double-check the numbers!
To determine if the water will reach the fire, we first need to find the horizontal distance the water will travel and compare it to the distance between the building and the fire.
Let's break down the given information:
Height of the building (10th floor) = 34.5 m
Water velocity = 29 m/s
Launch angle = 63°
Height at which the hose is held = 0.90 m
First, we need to find the time it takes for the water to reach the highest point of its trajectory. To do this, we'll use the equation for vertical motion:
vₑ = v₀ + at
Where:
vₑ = final velocity = 0 m/s (at the highest point)
v₀ = initial velocity = 29 m/s
a = acceleration = -9.8 m/s² (due to gravity)
t = time
Rearranging the equation:
t = (vₑ - v₀) / a
t = (0 - 29) / -9.8
t ≈ 2.96 s
Now, we can find the vertical distance traveled by the water using the equation:
Δy = v₀y * t + (1/2) * a * t²
Where:
v₀y = vertical component of initial velocity = v₀ * sinθ
Δy = vertical distance traveled
v₀y = 29 * sin(63°)
v₀y ≈ 25.54 m/s
Δy = (25.54 * 2.96) + (1/2) * (-9.8) * (2.96)²
Δy ≈ 37.36 m
Next, we'll find the horizontal distance traveled by the water using the equation:
Δx = v₀x * t
Where:
v₀x = horizontal component of initial velocity = v₀ * cosθ
Δx = horizontal distance traveled
v₀x = 29 * cos(63°)
v₀x ≈ 13.36 m/s
Δx = 13.36 * 2.96
Δx ≈ 39.56 m
Now, we can calculate the total distance between the building and the fire:
Total distance = horizontal distance (Δx) + height from the ground (0.90 m)
Total distance = 39.56 + 0.90
Total distance ≈ 40.46 m
Since the total distance is greater than the height of the building (34.5 m), we can conclude that the water will reach the fire.
To find the distance from the building where the hose should be held in order to put out the fire, we need to calculate the horizontal distance the water will travel:
Distance from building = horizontal distance (Δx)
Distance from building ≈ 39.56 m
Therefore, the hose should be held approximately 39.56 meters away from the building in order to put out the fire.
To determine whether the water from the fire hose will reach the fire on the 10th floor of the building, we need to consider the horizontal and vertical components of its velocity.
First, let's calculate the horizontal component of the water's velocity. We can use the given angle and the speed of the water:
Horizontal velocity (Vx) = velocity × cos(angle)
= 29 m/s × cos(63°)
≈ 29 m/s × 0.447
≈ 12.963 m/s
Next, let's calculate the time it takes for the water to reach the building. Since the horizontal velocity remains constant throughout, we can use the horizontal distance to determine the time. Assuming the fire truck is located at the origin (0,0), the horizontal distance is the position of the building:
Horizontal distance = 0.90 m
To calculate the time, we can use the equation:
time = distance / velocity
time = 0.90 m / 12.963 m/s
≈ 0.0695 s
Now, let's calculate the vertical distance the water travels during this time. We can use the equation of motion:
Vertical distance = initial vertical velocity × time + (0.5 × acceleration × time^2)
Since the water starts from the same height as the building, the initial vertical velocity is zero. The acceleration due to gravity is -9.8 m/s^2 (negative because it acts downward).
Vertical distance = 0 + (0.5 × (-9.8 m/s^2) × (0.0695 s)^2)
= 0 + (-0.0337 m)
≈ -0.034 m
The negative sign indicates that the water is lower than the original height.
Therefore, based on these calculations, we can conclude that the water from the fire hose will not reach the fire on the 10th floor of the building. It falls short by approximately 0.034 meters.
To put out the fire, the hose should be located closer to the building. The distance can be determined by considering where the water would reach the desired height. In this case, the desired height would be the height of the building (34.5 meters).
We can set up the vertical distance equation again, but this time solving for the initial vertical velocity since we know the vertical distance and time:
Vertical distance = initial vertical velocity × time + (0.5 × acceleration × time^2)
34.5 m = initial vertical velocity × time + (0.5 × (-9.8 m/s^2) × (time)^2)
This equation is quadratic and can be solved to find the initial vertical velocity using appropriate methods such as factoring, completing the square, or using the quadratic formula.
Once we find the initial vertical velocity, we can then calculate the horizontal distance using the time taken for the water to reach that height. The horizontal distance will give us the correct position for the hose to put out the fire.
Keep in mind that the calculations and final position mentioned above are based on the given information. In reality, firefighting involves various factors and variables, so it's important to consult professionals in emergency situations.