An airplane is flying with a velocity of v0 = 260 m/s at an angle of 30.0° with the horizontal. When the altitude of the plane is h = 2.0 km, a flare is released from the plane. The flare hits the target on the ground. What is the angle ?
To find the angle at which the flare hits the target on the ground, we can use the projectile motion equations.
First, we need to break down the initial velocity of the plane into its horizontal and vertical components. The horizontal component remains constant throughout the flight, while the vertical component changes due to gravity.
The horizontal component, vx, can be found using the formula:
vx = v0 * cos(theta)
where v0 is the initial velocity of the plane, and theta is the angle with the horizontal (given as 30.0°).
Substituting the given values:
vx = 260 m/s * cos(30.0°) ≈ 225.09 m/s
The vertical component, vy, can be found using the formula:
vy = v0 * sin(theta)
Substituting the given values:
vy = 260 m/s * sin(30.0°) ≈ 130 m/s
Next, we need to calculate the time it takes for the flare to reach the ground. We can use the formula:
h = vy * t + (1/2) * g * t^2
where h is the altitude of the plane (2.0 km or 2000 m), vy is the vertical component of the velocity, t is the time, and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Substituting the given values:
2000 m = (130 m/s) * t + (1/2) * (9.8 m/s^2) * t^2
This equation is a quadratic equation in terms of t. By solving it, we can find the time it takes for the flare to reach the ground. We can use the quadratic formula to solve for t:
t = (-b ± sqrt(b^2 - 4ac)) / (2a)
where a = (1/2) * g, b = vy, and c = -h.
Plugging in the values:
t = (-(130 m/s) ± sqrt((130 m/s)^2 - 4 * (0.5 * 9.8 m/s^2) * (-2000 m))) / (2 * 0.5 * 9.8 m/s^2)
Simplifying the equation gives us two possible values for t, one positive and one negative. Since time cannot be negative in this context, we can ignore the negative solution:
t ≈ 20.19 s
Finally, we can find the horizontal distance traveled by the flare using the formula:
dx = vx * t
Substituting the given values:
dx = (225.09 m/s) * (20.19 s) ≈ 4546.46 m
Since we now have the vertical and horizontal distances traveled, we can find the angle at which the flare hits the target using the tangent function:
tan(angle) = (vertical distance / horizontal distance)
angle = atan(vertical distance / horizontal distance)
angle ≈ atan(2000 m / 4546.46 m) ≈ 24.83°
Therefore, the angle at which the flare hits the target on the ground is approximately 24.83°.