A brick is thrown upward from the top of a building at an angle of 33.5◦ above the hori- zontal and with an initial speed of 12.8 m/s. The acceleration of gravity is 9.8 m/s2 . If the brick is in flight for 3.4 s, how tall is the building?
Vo = 12.8 m/s @ 33.5o
Yo = 12.8*sin33.5 = 7.06 m/s.
Tr = -(Yo)/g = -7.06/-9.8 = 0.721 s. =
Rise time.
Tf1 = Tr = 0.721 s. = Fall time from max
ht. to top of building.
Tr+Tf1+Tf2 = 3.4 s.
0.721 + 0.721 + Tf2 = 3.4
Tf2 = 3.4 - 1.44 = 1.96 s. = Fall time
from top of building to gnd.
h = Yo*Tf2 + 0.5g*Tf2^2
h = 7.06*1.96 + 4.9*1.96^2 = 32.7 m.
To find the height of the building, we need to determine the vertical displacement of the brick during its flight.
First, let's break down the motion of the brick into vertical and horizontal components. We can use the equations of motion separately for each component.
Vertical Component:
The initial vertical velocity (initial speed in the upward direction) can be found using trigonometry:
Vertical Velocity (Vy0) = Initial speed * sin(angle)
Vy0 = 12.8 m/s * sin(33.5°)
The vertical acceleration (due to gravity) is always -9.8 m/s^2 (negative since it acts downward).
Using the equation of motion, we can find the vertical displacement (height) of the brick:
Vertical Displacement (height) = (Vertical Velocity)^2 / (2 * Vertical Acceleration)
height = (Vy0^2) / (2 * (-9.8 m/s^2))
Horizontal Component:
The horizontal velocity (Vx) remains constant during the motion because there is no horizontal acceleration.
Horizontal Velocity (Vx) = Initial speed * cos(angle)
Vx = 12.8 m/s * cos(33.5°)
Since there is no horizontal acceleration, the horizontal displacement (distance traveled) can be found using the equation:
Horizontal Displacement (distance) = Horizontal Velocity * Time
distance = Vx * 3.4 s
The height of the building is given by the vertical displacement. Now we have all the values needed to calculate it:
1. Calculate the vertical velocity (Vy0):
Vy0 = 12.8 m/s * sin(33.5°)
2. Calculate the vertical displacement (height):
height = (Vy0^2) / (2 * (-9.8 m/s^2))
3. Calculate the horizontal velocity (Vx):
Vx = 12.8 m/s * cos(33.5°)
4. Calculate the horizontal displacement (distance):
distance = Vx * 3.4 s
Finally, to find the height of the building, you need to subtract the initial height of the brick from the vertical displacement (height).
Remember to consider the direction of the displacement (upward as positive or downward as negative), depending on the reference point for the height measurement.