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Chapter 3: Problem 77

A firefighter, \(60 \mathrm{~m}\) away from a burning building, directs a stream of water from a ground-level fire hose at an angle of \(37^{\circ}\) above the horizontal. If the water leaves the hose at \(40.3 \mathrm{~m} / \mathrm{s}\), which floor of the building will the stream of water strike? Each floor is \(4 \mathrm{~m}\) high.

Short Answer

Expert verified
Based on the given information, the stream of water will strike the 6th floor of the building.

Step by step solution

01

Identify known values

We know the following information: - The horizontal distance between the firefighter and the building \(d = 60\,\text{m}\) - The angle the water is being sprayed above the horizontal, \(\theta = 37^{\circ}\) - The speed the water leaves the hose, \(v_0 = 40.3\,\text{m/s}\) - The height of each floor, \(h_f = 4\,\text{m}\)
02

Find the horizontal and vertical components of the initial speed

We can use trigonometry to find the horizontal and vertical components of the initial speed. \(v_{0x} = v_0 \cos(\theta) = 40.3\,\text{m/s} \cos(37^{\circ}) \approx 32.2\,\text{m/s}\) \(v_{0y} = v_0 \sin(\theta) = 40.3\,\text{m/s} \sin(37^{\circ}) \approx 24.4\,\text{m/s}\)
03

Calculate the time it takes for the water to reach the building

We will use the horizontal component of the speed \(v_{0x}\) and the horizontal distance \(d\) to find the time it takes for the water to reach the building. \(d = v_{0x}t \Rightarrow t = \dfrac{d}{v_{0x}} = \dfrac{60\,\text{m}}{32.2\,\text{m/s}} \approx 1.86\,\text{s}\)
04

Calculate the vertical position of the water when it reaches the building

We will use the vertical component of the speed \(v_{0y}\), the time \(t\) we found in step 3, and the following equation of motion to determine the height the water reaches when it hits the building: \(h = v_{0y}t - \dfrac{1}{2}gt^2\) Where \(g = 9.81\,\text{m/s}^2\) is the acceleration due to gravity. \(h = (24.4\,\text{m/s})(1.86\,\text{s}) - \dfrac{1}{2}(9.81\,\text{m/s}^2)(1.86\,\text{s})^2 \approx 21.2\,\text{m}\)
05

Determine which floor the stream of water will strike

Using the height of each floor \(h_f\), divide the height \(h\) by \(h_f\) and round up to the nearest whole floor to find which floor the stream of water will strike. Number of Floors = \(\lceil \dfrac{h}{h_f} \rceil = \lceil \dfrac{21.2\,\text{m}}{4\,\text{m}} \rceil = 6\) The stream of water will strike the 6th floor of the building.

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