A golf ball is driven with the initial conditions shown in...

A golf ball is driven with the initial conditions shown in the figure. If the wind imparts a constant horizontal deceleration of $4 \mathrm{ft} / \mathrm{sec}^{2},$ determine the values of $r, \dot{r}, \ddot{r}, \theta, \dot{\theta},$ and $\ddot{\theta}$ when $t=1.05$ sec. Take the $r$ -coordinate to be measured from the origin.

Key Concepts

Effects of External Forces on Projectile Motion

When additional forces, such as wind, influence a projectile, they modify the standard kinematic equations. An understanding of how constant external accelerations affect the motion allows one to incorporate these forces into the analysis by adjusting the corresponding components of the acceleration in the coordinate system being used.

Decomposition of Acceleration

In polar coordinates, the total acceleration is broken down into a radial component (arising from changes in the magnitude of the radius and centripetal corrections) and a transverse component (due to angular acceleration). This decomposition is essential for separating the effects of forces acting towards or away from the origin from those causing rotational change.

Polar Coordinate System

This concept involves representing points and vectors through a distance (r) from a fixed origin and an angle (?) relative to a reference direction. It is widely used for problems where the geometry or motion naturally aligns with a central point or symmetry, such as in circular or orbital motion.

Kinematics in Polar Coordinates

Kinematics in polar coordinates involves analyzing motion by decomposing the position, velocity, and acceleration into radial and angular components. The derivatives of r and ? with respect to time (i.e., ?, r?, ??, and ??) describe how the distance from the origin and the direction change over time, which is crucial for understanding dynamics in non-Cartesian frameworks.

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