A brick is thrown upward from the top of a building at an angle of 15° to the horizontal and with an initial speed of 16 m/s. If the brick is in flight for 2.8 s, how tall is the building?

To solve this problem, we need to break it down into steps. Here's a step-by-step solution:

Step 1: Split the initial velocity into horizontal and vertical components
The initial velocity (16 m/s) can be split into horizontal and vertical components using trigonometry. The horizontal component (Vx) remains constant, while the vertical component (Vy) changes due to gravity.

Vx = V * cos(15°)
Vy = V * sin(15°)

where
V = initial speed = 16 m/s

Plugging in the numbers, we get:
Vx = 16 * cos(15°)
Vy = 16 * sin(15°)

Step 2: Calculate the time it takes for the brick to reach its highest point
The time it takes for the brick to reach its highest point (t1) can be calculated using the equation:
Vy = V0y - g * t1

where
g = acceleration due to gravity = 9.8 m/s^2

Rearranging the equation, we get:
t1 = (Vy - V0y) / g

Plugging in the numbers, we get:
t1 = (0 - 16 * sin(15°)) / -9.8

Step 3: Calculate the maximum height reached by the brick
The maximum height reached by the brick (h) can be calculated using the equation:
h = V0y * t1 + (1/2) * (-g) * t1^2

Plugging in the numbers, we get:
h = (16 * sin(15°)) * [(0 - 16 * sin(15°)) / -9.8] + (1/2) * (-9.8) * [((0 - 16 * sin(15°)) / -9.8)^2]

Step 4: Calculate the time it takes for the brick to fall back to the ground
The time it takes for the brick to fall back to the ground (t2) can be calculated using the equation:
2h = (1/2) * g * t2^2

Rearranging the equation, we get:
t2 = sqrt((2h) / g)

Plugging in the number for h calculated in Step 3, we get:
t2 = sqrt((2 * h) / 9.8)

Step 5: Calculate the total time of flight
The total time of flight (t_total) can be calculated by adding the time it takes to reach the maximum height (t1) and the time it takes to fall back to the ground (t2).
t_total = t1 + t2

Plugging in the numbers, we get:
t_total = [(0 - 16 * sin(15°)) / -9.8] + sqrt((2 * h) / 9.8)

Step 6: Calculate the height of the building
The height of the building (H) is equal to the maximum height reached by the brick (h). H = h

Plugging in the number for h calculated in Step 3, we get:
H = h

Now you can plug the values into your calculator to find the height of the building.

To determine the height of the building, we need to analyze the vertical motion of the brick.

First, we need to break down the initial velocity of the brick into its vertical and horizontal components. The initial speed of 16 m/s can be divided into two parts:

Vertical component: V_vertical = initial speed * sin(angle)
Horizontal component: V_horizontal = initial speed * cos(angle)

Given that the angle is 15°, we can calculate the components as follows:

V_vertical = 16 m/s * sin(15°) = 4.17 m/s (rounded to two decimal places)
V_horizontal = 16 m/s * cos(15°) = 15.32 m/s (rounded to two decimal places)

Next, we can determine the time it takes for the brick to reach its maximum height. Since the brick is thrown upwards, its vertical velocity will decrease until it reaches zero at the maximum height. We can use the formula:

V_vertical_final = V_vertical_initial - g * t

Where:
V_vertical_final = 0 m/s (at maximum height)
V_vertical_initial = 4.17 m/s (from earlier calculation)
g = acceleration due to gravity = 9.8 m/s^2
t = time to reach maximum height (unknown)

Rearranging the equation, we get:

t = (V_vertical_initial - V_vertical_final) / g

Plugging in the known values:

t = (4.17 m/s - 0 m/s) / 9.8 m/s^2 = 0.426 s (rounded to three decimal places)

Since the total time of flight is 2.8 s, we can determine the time it takes for the brick to fall back to its initial height:

Total time = time to reach maximum height + time to fall back

2.8 s = 0.426 s + time to fall back

Rearranging the equation, we get:

time to fall back = 2.8 s - 0.426 s = 2.374 s (rounded to three decimal places)

We know that the time to fall back is twice the time it took to reach the maximum height since the motion is symmetrical. Therefore:

time to reach maximum height = (time to fall back) / 2 = 2.374 s / 2 = 1.187 s (rounded to three decimal places)

Now, we can calculate the maximum height reached by the brick using the formula:

height = V_vertical_initial * t - 0.5 * g * t^2

Where:
V_vertical_initial = 4.17 m/s (from earlier calculation)
g = acceleration due to gravity = 9.8 m/s^2
t = time to reach maximum height (1.187 s, from earlier calculation)

Plugging in the values:

height = 4.17 m/s * 1.187 s - 0.5 * 9.8 m/s^2 * (1.187 s)^2 = 9.913 m (rounded to three decimal places)

Therefore, the height of the building is approximately 9.913 meters.