A brick is thrown upward from the top of a building at an angle of 25° to the horizontal and with an initial speed of 16 m/s. If the brick is in flight for 3.1 s, how tall is the building?
The vertical component of the speed is initially
Voy = 16 sin25 = 6.762 m/s.
Height above ground is
y = H + Voy*t - (g/2)t^2
y = H +6.762t -4.9 t^2
You know that y = 0 when t = 3.1 s.
Solve for H, the building height.
To find the height of the building, we need to determine the vertical component of the brick's motion.
Step 1: Find the initial vertical velocity (Vy) of the brick.
Given:
Initial speed (V0) = 16 m/s
Launch angle (θ) = 25°
The initial vertical velocity can be calculated using the formula:
Vy = V0 * sin(θ)
Substituting the given values:
Vy = 16 m/s * sin(25°)
Vy ≈ 6.79 m/s
Step 2: Find the time taken to reach the highest point (t1).
Since the brick is thrown upward, it will reach its highest point when its vertical velocity becomes zero.
The time taken to reach the highest point, t1, can be calculated using the formula:
t1 = Vy / g
Where g is the acceleration due to gravity (approximately 9.8 m/s²).
Substituting the values:
t1 = 6.79 m/s / 9.8 m/s²
t1 ≈ 0.69 s
Step 3: Find the time taken to reach the ground (t2).
The total time of flight is given as 3.1 s, and we know that the time taken to reach the highest point is 0.69 s.
Thus, the time taken to reach the ground, t2, is:
t2 = Total time of flight - Time taken to reach the highest point
t2 = 3.1 s - 0.69 s
t2 ≈ 2.41 s
Step 4: Find the vertical displacement (h) using the time taken to reach the ground (t2).
Using the formula for displacement with constant acceleration:
h = Vy * t2 - 0.5 * g * t2²
Substituting the known values:
h = 6.79 m/s * 2.41 s - 0.5 * 9.8 m/s² * (2.41 s)²
h ≈ 16.33 m
Therefore, the height of the building is approximately 16.33 meters.
To solve this problem, we can break it down into two parts: the horizontal motion and the vertical motion of the brick.
1. Horizontal Motion:
The initial speed of the brick along the horizontal direction does not change throughout the flight. Therefore, we can use the equation:
Horizontal displacement = initial speed * time
Horizontal displacement = 16 m/s * 3.1 s
2. Vertical Motion:
We can analyze the vertical motion of the brick using the equations of motion. The key equation to use in this case is:
Vertical displacement = initial vertical velocity * time + (1/2) * acceleration * time^2
To calculate the initial vertical velocity, we use the given information that the brick is thrown at an angle of 25° to the horizontal. We can use trigonometry to find the vertical component of the initial velocity:
Initial vertical velocity = initial speed * sin(angle)
Now we can substitute the known values into the equation above to find the vertical displacement.
Once we have the horizontal and vertical displacements, we can calculate the total displacement, which will give us the height of the building.
Now, let's calculate the values step by step:
1. Horizontal Motion:
Horizontal displacement = 16 m/s * 3.1 s = 49.6 m
2. Vertical Motion:
Initial vertical velocity = 16 m/s * sin(25°)
Now we need to calculate the acceleration due to gravity to use it in the vertical motion equation. Assuming no air resistance, the acceleration due to gravity is approximately 9.8 m/s^2.
Vertical displacement = (16 m/s * sin(25°)) * 3.1 s + (1/2) * 9.8 m/s^2 * (3.1 s)^2
Finally, we can find the total displacement (height of the building) by combining the horizontal and vertical displacements.
Total displacement = sqrt((horizontal displacement)^2 + (vertical displacement)^2)
Therefore, the height of the building can be calculated using the above steps.