An airplane is flying with a velocity of 252m/s at an angle of 30 degrees with the horizontal. when the altitude of the plane is 2.7km, a flare is released from the plane. the flare hits the target on the ground. what is the angle theta?
To find the angle theta, we need to use trigonometry. Let's break down the problem into components:
1. The horizontal component of the velocity (Vx) can be found using the formula Vx = V * cos(theta), where V is the given velocity (252 m/s) and theta is the given angle (30 degrees).
Vx = 252 m/s * cos(30 degrees)
= 252 m/s * √3/2
≈ 218.35 m/s
2. The time it takes for the flare to hit the ground can be found using the formula t = d / Vx, where d is the altitude of the plane (2.7 km = 2700 m) and Vx is the horizontal component of the velocity.
t = 2700 m / 218.35 m/s
≈ 12.36 s
3. The vertical component of the velocity (Vy) can be found using the formula Vy = V * sin(theta), where V is the given velocity (252 m/s) and theta is the given angle (30 degrees).
Vy = 252 m/s * sin(30 degrees)
= 252 m/s * 1/2
= 126 m/s
4. The maximum height (H) reached by the flare can be found using the kinematic equation: H = Vy * t - (1/2) * g * t^2, where g is the acceleration due to gravity (9.8 m/s^2) and t is the time calculated earlier.
H = 126 m/s * 12.36 s - (1/2) * 9.8 m/s^2 * (12.36 s)^2
≈ 3863.42 m
5. The angle theta can be found using the inverse tangent function: theta = arctan(H / d), where H is the maximum height reached by the flare and d is the altitude of the plane.
theta = arctan(3863.42 m / 2700 m)
≈ 56.3 degrees
Therefore, the angle theta is approximately 56.3 degrees.