An airplane is flying with a velocity of 240 m/s at an angle of 30.0° with the horizontal, as the drawing shows. When the altitude of the plane is 2.2 km, a flare is released from the plane. The flare hits the target on the ground. What is the angle θ?
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To find the angle θ, we need to break down the velocity vector of the airplane into its horizontal and vertical components.
Given that the velocity of the airplane is 240 m/s at an angle of 30.0° with the horizontal, we can calculate the horizontal and vertical components of this velocity.
Horizontal component (Vx) = Velocity * cos(angle)
Vertical component (Vy) = Velocity * sin(angle)
Vx = 240 m/s * cos(30.0°) ≈ 207.9 m/s
Vy = 240 m/s * sin(30.0°) ≈ 120 m/s
Now, let's consider the motion of the flare once it is released from the plane. Since there are no horizontal forces acting on the flare (neglecting air resistance), the horizontal component of the flare's velocity remains constant throughout its trajectory. Therefore, the horizontal component (Vx_flare) of the velocity of the flare is the same as the horizontal component of the airplane's velocity.
Vx_flare = 207.9 m/s
However, the vertical component (Vy_flare) of the velocity of the flare changes since it is affected by gravitational acceleration. The vertical component of the velocity initially is the same as the vertical component of the airplane's velocity (Vy_flare = 120 m/s), but it decreases due to the acceleration due to gravity (-9.8 m/s²).
Now, let's determine the time it takes for the flare to hit the ground. We can use the equation of motion for vertical motion:
Vy_flare = Vy + (Acceleration due to gravity * time)
Since Vy_flare = 0 when the flare hits the ground, we can rearrange the equation to solve for time:
0 = 120 m/s - (9.8 m/s² * time)
120 m/s = 9.8 m/s² * time
time = 120 m/s / 9.8 m/s² ≈ 12.24 s
Next, we can use this time (12.24 s) to find the horizontal distance traveled by the flare.
Distance = Velocity * time
Distance = Vx_flare * time
Distance = 207.9 m/s * 12.24 s ≈ 2543.2 m
Now, we have the horizontal distance traveled by the flare, which is the same as the horizontal distance between the plane and the target on the ground. So, we can use the tangent function to find the angle θ.
tan(θ) = opposite / adjacent
tan(θ) = 2.2 km / 2543.2 m
θ = arctan(2.2 km / 2543.2 m)
Evaluating this using a calculator, we get:
θ ≈ 40.5°
Therefore, the angle θ is approximately 40.5°.