Instruments located at O are part of the groundtraffic...

Instruments located at $O$ are part of the groundtraffic control system for a major airport. At a certain instant during the takeoff roll of the aircraft $P,$ the sensors indicate the angle $\theta=50^{\circ}$ and the range rate $\dot{r}=140 \mathrm{ft} / \mathrm{sec} .$ Determine the corre sponding speed $v$ of the aircraft and the value of $\dot{\theta}$

Key Concepts

Polar Coordinate System

The polar coordinate system represents a point in the plane using two parameters: the distance r from a fixed origin and the angle ? measured from a fixed reference direction. This system is especially useful when describing problems involving rotations, radial motions, or systems with circular symmetry, as it simplifies the geometric interpretation of movement along a curve.

Differentiation in Polar Coordinates

In polar coordinates, when the position of a point is given by (r, ?), taking time derivatives involves both the rate at which the distance changes (dr/dt) and the rate at which the angle changes (d?/dt). Differentiation using the chain rule allows one to determine how the overall position changes in time and subsequently how the various components of motion—radial and tangential—contribute to the speed.

Range Rate

The range rate refers to the time derivative of the distance between a moving object and a fixed point; in other words, it is dr/dt. This quantity measures how quickly an object is approaching or receding from the origin and is central in problems where only the radial component of motion is directly measurable.

Angular Rate

The angular rate, d?/dt, is the rate at which the angle ? changes with respect to time. It quantifies the rotational aspect of the motion, indicating how quickly the line connecting the object to the origin is rotating, and is key in breaking down the velocity into its polar components.

Velocity Vector Decomposition in Polar Coordinates

When an object moves in the plane, its overall speed (or magnitude of velocity) can be computed by decomposing the velocity vector into two orthogonal components: the radial component (dr/dt) and the tangential component (r d?/dt). The combination of these components through the Pythagorean theorem—v^2 = (dr/dt)^2 + (r d?/dt)^2—provides the complete picture of the object's motion.

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