A meteor P is tracked by a radar observatory on the earth at...

A meteor $P$ is tracked by a radar observatory on the earth at $O .$ When the meteor is directly overhead $\left(\theta=90^{\circ}\right),$ the following observations are recorded: $r=80 \mathrm{km}, \dot{r}=-20 \mathrm{km} / \mathrm{s},$ and $\dot{\theta}=0.4$ $\operatorname{rad} / \mathrm{s}$ (a) Determine the speed $v$ of the meteor and the angle $\beta$ which its velocity vector makes with the horizontal. Neglect any effects due to the earth's rotation. (b) Repeat with all given quantities remaining the same, except that $\theta=75^{\circ}$

A meteor $P$ is tracked by a radar observatory on the earth at $O .$ When the meteor is directly overhead $\left(\theta=90^{\circ}\right),$ the following observations are recorded: $r=80 \mathrm{km}, \dot{r}=-20 \mathrm{km} / \mathrm{s},$ and $\dot{\theta}=0.4$
$\operatorname{rad} / \mathrm{s}$
(a) Determine the speed $v$ of the meteor and the angle $\beta$ which its velocity vector makes with the horizontal. Neglect any effects due to the earth's rotation.
(b) Repeat with all given quantities remaining the same, except that $\theta=75^{\circ}$

Key Concepts

Angle Determination in Polar Coordinates

The direction of the velocity vector relative to a reference direction (often the horizontal) is determined by the arctan of the ratio of the angular component to the radial component. This process converts the polar components into an effective angle that shows the orientation of the total velocity vector, thereby helping to visualize both the direction and the magnitude of the motion.

Vector Magnitude Calculation

The total speed of an object moving in a plane is obtained by combining its radial and angular velocity components using the Pythagorean theorem. This magnitude is calculated as the square root of the sum of the squares of the radial velocity and the product of the radius with the angular velocity. This approach allows for a clear understanding of the resulting motion from the contributions of both directions.

Polar Coordinate System

The polar coordinate system represents points in a plane by a distance from a fixed origin and an angle from a fixed direction. This coordinate system is particularly useful in problems involving radial motion or rotations, where expressing positions and velocities in terms of a radius and angle simplifies the analysis of the problem.

Radial and Angular Components of Velocity

In polar coordinates, the velocity of an object is decomposed into two components: a radial component (dot{r}) which measures the rate of change of the distance from the origin, and an angular component (r dot{theta}) which measures the rate of change of the angle. These components are key to understanding the object's motion as they indicate how fast it is moving directly away from or towards the origin, and how fast it is sweeping out angular distance.

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