For the helicopter of Prob. 2 / 172, find the values of R̈, θ̈, and ϕ̈ for the radar tracking de

step 1 In this question it is told that a ball is dropped vertically from a height of the ground and it

For the helicopter of Prob. 2 / 172, find the values of R̈,...

For the helicopter of Prob. $2 / 172,$ find the values of $\ddot{R}, \ddot{\theta},$ and $\ddot{\phi}$ for the radar tracking device at $O$ at the instant when $h=100 \mathrm{m} .$ Refer to the printed answers for Prob. $2 / 172$ as needed.

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Key Concepts

Acceleration Components in Spherical Coordinates

The acceleration in spherical coordinates comprises several components: one in the radial direction and others associated with the angular motion. Deriving these components involves computing second time derivatives of the radial and angular variables, along with additional terms that arise from the coupling between these parameters. This breakdown is essential for accurately analyzing the motion of objects that are being tracked or observed from a fixed point.

Time Differentiation in Non?Cartesian Coordinates

When a point is expressed in a non-Cartesian framework like spherical coordinates, its basis vectors vary with position and time. As a result, forming time derivatives (like velocities and accelerations) requires not only differentiating the coordinate magnitudes but also accounting for the changes in the direction of the basis vectors.

Spherical Coordinate System

A spherical coordinate system represents points in three-dimensional space using a radius and two angles. This system is especially useful in problems with rotational symmetry and when describing motions that emanate from or converge towards a point, such as the path of an object relative to a fixed tracking device.

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