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

A helicopter starts from rest at point $A$ and travels along the straight-line path with a constant acceleration $a$. If the speed $v=28 \mathrm{m} / \mathrm{s}$ when the altitude of the helicopter is $h=40 \mathrm{m},$ determine the values of $\dot{r}, \ddot{r}, \dot{\theta},$ and $\ddot{\theta}$ as measured by the tracking device at $O .$ At this instant, $\theta=40^{\circ},$ and the distance $d=160 \mathrm{m} .$ Neglect the small height of the tracking device above the ground.

Key Concepts

Time Differentiation in Polar Coordinates

Differentiating vectors in polar coordinates requires careful application of the chain rule because both the radial coordinate and the angular coordinate are functions of time. This yields expressions for the radial acceleration (involving r? and the square of the angular speed) and the angular acceleration (including terms that account for changes in both r and ??), which explain how the motion evolves relative to the fixed point.

Relative Motion and Coordinate Transformation

When an object is tracked from a fixed reference point, it is necessary to transform its motion from its natural path description (often in Cartesian coordinates) to the observer’s polar coordinate system. This involves establishing geometric relationships between the object’s real trajectory—such as its known speed and altitude—and its observed motion in terms of radial distance and angle, allowing for precise determination of kinematic quantities like r?, r?, ??, and ??.

Radial and Angular Components of Kinematics

In polar coordinates, any velocity or acceleration vector is split into radial components (relating to changes in the distance from the origin) and angular components (relating to changes in the angle). The velocity is expressed as r? in the radial direction and r?? in the angular direction, while the acceleration includes contributions from both the direct rate of change of the radial speed and additional terms involving the angular speed and its rate of change.

Polar Coordinate System

This is a method of describing a point in the plane by its distance from a fixed central point (the origin) and the angle from a fixed reference direction. It is particularly useful when analyzing motions observed from a fixed point, as it naturally decomposes the position, velocity, and acceleration into radial and angular components.

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