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 2.4 km, a flare is released from the plane. The flare hits the target on the ground. What is the angle è?
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give up...its been 6 years...
To find the angle θ, we need to use trigonometry. Let's break down the given information:
Velocity of the airplane (V) = 240 m/s
Angle of the airplane with the horizontal (α) = 30.0°
Altitude of the plane (h) = 2.4 km = 2400 m
Using the given information, we can find the horizontal and vertical components of the velocity. The horizontal component (Vx) is given by:
Vx = V * cos(α)
Plugging in the values, we get:
Vx = 240 m/s * cos(30.0°) = 240 m/s * 0.866 = 207.84 m/s
The vertical component (Vy) is given by:
Vy = V * sin(α)
Plugging in the values, we get:
Vy = 240 m/s * sin(30.0°) = 240 m/s * 0.5 = 120 m/s
Now, we can use the vertical motion equation to find the time (t) it takes for the flare to hit the ground:
h = Vy * t + (1/2) * g * t^2
Where g is the acceleration due to gravity (-9.8 m/s²).
Plugging in the values, we get:
2400 m = 120 m/s * t + (1/2) * (-9.8 m/s²) * t^2
Rearranging the equation, we get:
4.9t^2 + 120t - 2400 = 0
Now, we can solve this quadratic equation for t. Using the quadratic formula, we get two values for t: approximately -19.12 s and 7.09 s. Since time cannot be negative in this case, we discard the negative value.
The time it takes for the flare to hit the ground is approximately 7.09 seconds.
Now, we can find the horizontal distance covered by the flare using the time t:
Horizontal distance (d) = Vx * t
Plugging in the values, we get:
d = 207.84 m/s * 7.09 s = 1473.96 m
Now, we can find the angle θ using the tangent function:
tan(θ) = h / d
Plugging in the values, we get:
tan(θ) = 2400 m / 1473.96 m
Calculating the value, we get:
θ ≈ tan^(-1)(1.627)
Using a calculator, we find:
θ ≈ 57.19°
Therefore, the angle θ is approximately 57.19°.
To find the angle è, we need to use the concept of projectile motion. The motion of the flare can be divided into horizontal and vertical components.
First, let's calculate the horizontal component of the velocity. The horizontal velocity of the airplane is 240 m/s, and we can find the horizontal component of the flare's velocity by multiplying it by the cosine of the angle 30.0°:
Horizontal velocity (Vx) = 240 m/s * cos(30.0°) = 240 m/s * 0.866 = 207.84 m/s
Now, let's calculate the time it takes for the flare to hit the ground. The vertical motion of the flare can be described by the equation:
y = Vyi * t - 0.5 * g * t²
where:
y = vertical distance traveled (2.4 km = 2400 m)
Vyi = initial vertical velocity of the flare (0 m/s, as it starts from rest)
g = acceleration due to gravity (-9.8 m/s²)
t = time
We can rearrange the equation to solve for t:
0 = 0 * t - 0.5 * (-9.8 m/s²) * t²
0 = 4.9 m/s² * t²
Solving for t, we get:
t² = 0
t = 0 seconds
Since t = 0, it means the flare hits the target instantly. Therefore, the angle è is 0°.