A fireman d = 39.0 m away from a burning building directs a stream of water from a ground-level fire hose at an angle of èi = 25.0° above the horizontal as shown in the figure. If the speed of the stream as it leaves the hose is vi = 40.0 m/s, at what height will the stream of water strike the building?
m
To find the height at which the stream of water will strike the building, we need to break down the stream's initial velocity into its horizontal and vertical components.
Given:
- Distance from the fireman to the building, d = 39.0 m
- Angle of the stream above the horizontal, θi = 25.0°
- Initial speed of the stream, vi = 40.0 m/s
First, we need to determine the horizontal and vertical components of the initial velocity.
Horizontal Component (Vx):
The horizontal component of the velocity remains constant throughout the motion since there are no horizontal forces acting on the water stream.
Vx = vi * cos(θi)
Vertical Component (Vy):
The vertical component of the velocity is affected by gravity.
Vy = vi * sin(θi) - (g * t)
Since the water stream hits the building at the highest point of its trajectory, it reaches its maximum height when its vertical component becomes zero (Vy = 0).
Now, set Vy = 0 and solve for time, t:
0 = vi * sin(θi) - (g * t)
t = vi * sin(θi) / g
Finally, substitute the time, t, obtained into the equation for the height, h, traveled vertically by the water stream:
h = (vi * sin(θi) * t) - (0.5 * g * t^2)
Plug in the given values and calculate h to find the height at which the stream of water will strike the building.