Answer:
To calculate the horizontal distance from the base of the building to the point where the rock strikes the ground, we can analyze the motion of the rock in the horizontal and vertical directions.
Given:
Height of the building (h) = 15.0 m
Magnitude of the velocity (v) = 24.0 m/s
Launch angle (θ) = 41.0 degrees
Acceleration due to gravity (g) = 9.8 m/s^2
Horizontal Motion:
The horizontal motion of the rock is unaffected by gravity. The horizontal velocity (vx) remains constant throughout the motion.
vx = v * cos(θ)
Vertical Motion:
The vertical motion of the rock is influenced by gravity. We can analyze it using the equations of motion.
Initial vertical velocity (uy) = v * sin(θ)
Time of flight (t) is the time it takes for the rock to reach the ground from the roof of the building.
Using the equation of motion:
h = uy * t - (1/2) * g * t^2
We need to solve this equation for time (t) to find the time of flight.
At the highest point of the trajectory, the vertical velocity becomes zero. We can find the time it takes to reach the highest point using the equation:
0 = uy - g * t_max
t_max = uy / g
The total flight time will be twice the time it takes to reach the highest point (as the rock takes the same amount of time to reach the highest point as it does to fall back to the ground):
Total flight time = 2 * t_max
Now we can calculate the horizontal distance from the base of the building to the point where the rock strikes the ground using the formula:
Horizontal distance = vx * t
Let's calculate the horizontal distance:
Substituting the given values:
h = 15.0 m
v = 24.0 m/s
θ = 41.0 degrees
g = 9.8 m/s^2
uy = v * sin(θ)
t_max = uy / g
Total flight time = 2 * t_max
Horizontal distance = vx * t
Calculating:
uy = (24.0 m/s) * sin(41.0 degrees)
uy ≈ 24.0 m/s * 0.656
uy ≈ 15.74 m/s
t_max = (15.74 m/s) / (9.8 m/s^2)
t_max ≈ 1.61 seconds
Total flight time = 2 * 1.61 seconds
Total flight time ≈ 3.22 seconds
vx = v * cos(θ)
vx = (24.0 m/s) * cos(41.0 degrees)
vx ≈ 24.0 m/s * 0.759
vx ≈ 18.216 m/s
Horizontal distance = (18.216 m/s) * (3.22 seconds)
Horizontal distance ≈ 58.69 meters
Therefore, the horizontal distance from the base of the building to the point where the rock strikes the ground is approximately 58.69 meters.