A helicopter starts from rest at point A and travels along...

A helicopter starts from rest at point $A$ and travels along the indicated path with a constant acceleration $a$. If the helicopter has a speed of $60 \mathrm{m} / \mathrm{s}$ when it reaches $B$, determine the values of $\dot{R}, \dot{\theta},$ and $\dot{\phi}$ as measured by the radar tracking device at $O$ at the instant when $h=100 \mathrm{m}$

Key Concepts

Spherical Coordinate System

This coordinate system expresses the position of an object using a distance from a fixed origin and two angles. It is crucial in radar tracking applications, where the range, azimuth, and elevation angle are used to describe the position of an object relative to the tracking station. Analyzing the problem in these coordinates allows one to break down the object’s motion into components that are directly measurable by the radar device.

Radial and Angular Velocities

In non?Cartesian frameworks like spherical coordinates, the velocity of an object is decomposed into a radial component (the rate of change of the distance from the origin) and angular components (the rates of change of the azimuthal and elevation angles). Calculating these derivatives is key to understanding how the object's motion projects onto the radar’s measurement axes and is essential for extracting physical information from the observed data.

Constant Acceleration Kinematics

Constant acceleration kinematics describes the motion of an object that starts from rest and accelerates uniformly. This principle allows one to relate the displacement, speed, and time elapsed, and to calculate intermediate velocities and positions. In the context of the problem, these equations are used to determine the helicopter’s speed and positional parameters at a given height.

Application of the Chain Rule in Differentiation

When working with time-dependent coordinate systems like spherical coordinates, the chain rule is applied to differentiate composite functions. It is necessary for relating the time derivatives of the position components (such as range and angles) to the overall motion of the object. This method is critical when calculating the instantaneous rates of change as measured by the radar tracking device.

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