A fireman, 29.1 m away from a burning building, directs a stream of water from a ground level fire hose at an angle of 48.5° above the horizontal. If the speed is 51.6 m/s, at what height will the stream of water hit the building?
Friction is a big factor in water streams in the air. Ignoring friction,
hf=hi+vi'*t-4.9t^2
where t can be found by the horizontal
29.1=vi" * t
where
vi' (vertical)=51.6*sin48.5
vi" (hoizontal)=51.6*cos48.5
To find the height at which the stream of water hits the building, we can break down the problem into two components: the horizontal component and the vertical component.
First, let's find the time it takes for the stream of water to reach the building. We can use the horizontal component of the water's speed. The horizontal speed of the water can be found using the given speed of 51.6 m/s and the angle of 48.5°.
Horizontal Speed = Speed * cos(Angle)
Horizontal Speed = 51.6 m/s * cos(48.5°)
Using a calculator, we can determine that the horizontal speed is approximately 34.21 m/s.
Next, we'll use this horizontal speed to find the time it takes for the water to reach the building, which is 29.1 m away.
Time = Distance / Horizontal Speed
Time = 29.1 m / 34.21 m/s
Calculating this, we find that the time it takes for the water to reach the building is approximately 0.850 seconds.
Now, let's focus on the vertical component of the water's motion. The water starts from the ground level, so its initial vertical height is 0 m. We need to determine the final vertical height at which the water hits the building.
Using the equation of motion for vertical projectile motion:
Final Vertical Height = Initial Vertical Height + (Vertical Component of Speed * Time) + (1/2 * Acceleration * Time^2)
In this case, the initial vertical height is 0 m, the vertical component of speed is the vertical component of the water's speed, and the acceleration is due to gravity, which is approximately -9.8 m/s^2.
The vertical component of the water's speed can be found using the given speed of 51.6 m/s and the angle of 48.5°.
Vertical Speed = Speed * sin(Angle)
Vertical Speed = 51.6 m/s * sin(48.5°)
Using a calculator, we find that the vertical speed is approximately 36.63 m/s.
Now, let's calculate the final vertical height.
Final Vertical Height = 0 m + (36.63 m/s * 0.850 s) + (1/2 * -9.8 m/s^2 * (0.850 s)^2)
Calculating this, we find that the final vertical height is approximately 19.67 meters.
Therefore, the stream of water will hit the building at a height of approximately 19.67 meters above the ground.