A certain airplane has a speed of 336.8 km/h and is diving at an angle of θ = 32.0° below the horizontal when the pilot releases a radar decoy (see the figure). The horizontal distance between the release point and the point where the decoy strikes the ground is d = 614 m. (a) How long is the decoy in the air? (b) How high was the release point?
To find the answers to the given problem, we can break it down into two components: horizontal motion and vertical motion. Let's start by analyzing the horizontal motion.
(a) How long is the decoy in the air?
1. Determine the horizontal component of the decoy's velocity:
The horizontal component of the velocity can be found using the formula:
Vx = V * cos(θ),
where V is the speed of the airplane and θ is the angle of descent. Substituting the given values:
Vx = 336.8 km/h * cos(32.0°).
Convert the given speed to meters per second (m/s) since the vertical distance is given in meters:
Vx = (336.8 km/h * 1000 m/km) / (3600 s/hour) * cos(32.0°).
Calculate the value of Vx:
Vx ≈ 214.383 m/s.
2. Calculate the time of flight:
The time of flight can be calculated using the formula:
t = d / Vx,
where d is the horizontal distance traveled by the decoy.
Substituting the given values:
t = 614 m / 214.383 m/s.
Calculate the value of t:
t ≈ 2.86 seconds.
Therefore, the decoy is in the air for approximately 2.86 seconds.
(b) How high was the release point?
To find the height of the release point, we need to analyze the vertical motion.
1. Determine the vertical component of the decoy's velocity:
The vertical component of the velocity can be found using the formula:
Vy = V * sin(θ),
where V is the speed of the airplane and θ is the angle of descent. Substituting the given values:
Vy = 336.8 km/h * sin(32.0°).
Convert the given speed to meters per second (m/s):
Vy = (336.8 km/h * 1000 m/km) / (3600 s/hour) * sin(32.0°).
Calculate the value of Vy:
Vy ≈ 182.846 m/s.
2. Calculate the height of the release point (h):
To find the height, we need to use the kinematic equation:
h = Vy * t - (1/2) * g * t^2,
where g is the acceleration due to gravity (approximately 9.8 m/s^2) and t is the time of flight.
Substituting the values:
h = (182.846 m/s * 2.86 s) - (1/2) * (9.8 m/s^2) * (2.86 s)^2.
Calculate the value of h:
h ≈ 448.189 m.
Therefore, the release point was approximately 448.189 meters high.