Question

In a scenario where an airplane is flying at a uniform speed v over a radar tracking station A and passes directly over it at an altitude h, what are the following values calculated in relation to the radar antenna:

1. Angular velocity (q) of the radar antenna.
2. Angular acceleration (€q) of the radar antenna.
3. The rate (r) at which the airplane is moving away from the antenna.

Provide the expressions for these values using the equations of the relevant physics chapter. Assume an inertial frame attached to the ground and a non-rotating Earth, with the moving frame attached to the antenna, having the x-axis directed from the antenna toward the airplane.

Answer 1

The motion of an airplane relative to a radar antenna involves calculations of angular velocity (q), which is the change in angle over time, and angular acceleration (€q), which is zero in uniform motion. The rate (r) at which the airplane moves away from the antenna is derived from the horizontal distance covered as a function of time.

Step-by-step explanation:

The question is examining the motion of an airplane in relation to a radar antenna and involves calculating the angular velocity, angular acceleration, and the rate at which the airplane is moving away from the antenna.

Angular Velocity (q)

Angular velocity (ω) can be calculated by the change in angle (Δθ) over the change in time (Δt). As the airplane moves directly over and past the antenna, it creates a right triangle with the altitude h and the distance it moves with respect to the antenna. The angular velocity is thus the rate of change of the angle between the line from the airplane to the antenna and the perpendicular to the ground.

Angular Acceleration (€q)

Angular acceleration (α) is the rate of change of angular velocity over time. In uniform motion with constant speed v, the angular acceleration is zero because the angular velocity is constant, assuming no other forces act on the airplane or radar antenna.

The Rate (r) at Which the Airplane is Moving Away from the Antenna

The rate (r) at which the airplane is moving away from the antenna can be found using the Pythagorean theorem in the context of the present scenario where r is a function of time as the airplane moves away horizontally at speed v from the point above the antenna.