A fireman, 43.7 m away from a burning building, directs a stream jet of water from a ground level fire hose at an angle of 31.3 degree above the horizontal.
If the speed of the stream as it leaves the hose is 44.1 m/s, at what height will the stream of water strike the building? The acceleration due to gravity is 9.8 m/s^.
Answer in units of m.
This problem is practically identical to the one I just showed you how to do with the two cliffs.
horizontal:
u = 44.1 cos 31.3 forever
t = 43.7/u
vertical
h = 0 + Vi t - 4.9 t^2
where Vi = 44.1 sin 31.3
To solve this problem, we need to break it down into its vertical and horizontal components. We will find the vertical and horizontal distances separately and then combine them to get the final answer.
First, let's find the vertical distance (height) at which the stream of water will strike the building.
Given:
Initial velocity of the stream, v0 = 44.1 m/s
Angle of projection, θ = 31.3 degrees
Acceleration due to gravity, g = 9.8 m/s^2
Using the equation for vertical displacement in projectile motion:
Displacement (Δy) = v0 * sin(θ) * t + (1/2) * g * t^2
We need to find the time (t) it takes for the water to reach the building. The horizontal distance traveled by the water can be used to find the time.
The horizontal distance (range), R = v0 * cos(θ) * t
Given that the fireman is 43.7 m away from the building, the horizontal distance is R = 43.7 m.
Using the equation for horizontal displacement in projectile motion, we can rearrange it to solve for time (t):
t = R / (v0 * cos(θ))
Plugging in the values:
t = 43.7 m / (44.1 m/s * cos(31.3 degrees))
t ≈ 1.297 seconds (rounded to three decimal places)
Now that we have the time, we can calculate the vertical distance (height) using the first equation:
Δy = v0 * sin(θ) * t + (1/2) * g * t^2
Plugging in the values:
Δy = 44.1 m/s * sin(31.3 degrees) * 1.297 seconds + (1/2) * 9.8 m/s^2 * (1.297 seconds)^2
Δy ≈ 15.58 meters (rounded to two decimal places)
Therefore, the stream of water will strike the building at a height of approximately 15.58 meters.