A car traveling on a straight highway at an initial speed of $v_0 = 25 \, \text{m/s}$ (approx. $90 \, \text{km/h}$) applies its brakes and decelerates uniformly at $a = -5 \, \text{m/s}^2$. Determine the time required for the car to come to a complete stop and the total stopping distance.

A radar station tracking a rocket records its distance $r = 10,000 \, \text{m}$ and angle $\theta = 30^\circ$ at a certain instant. The tracking data indicates that $\dot{r} = 500 \, \text{m/s}$, $\ddot{r} = 20 \, \text{m/s}^2$, $\dot{\theta} = 0.05 \, \text{rad/s}$, and $\ddot{\theta} = -0.01 \, \text{rad/s}^2$. Determine the velocity and acceleration vectors of the rocket in terms of their radial and transverse components.

A projectile is fired from the ground with an initial velocity of $$ at an angle of $40^\circ$ with the horizontal. Determine the maximum height reached and the horizontal range of the projectile.

A race car is traveling along a circular curve of radius $$. At a specific instant, its speed is $$ and the driver is pressing the gas pedal, increasing the speed at a constant rate of $$. Determine the magnitude of the total acceleration of the car.

Two identical satellites, A and B, are orbiting the Earth in circular paths. Satellite A is in a low Earth orbit (LEO), while Satellite B is in a much higher geosynchronous orbit (GEO). The central force (gravity) acting on both satellites provides the normal acceleration required for circular motion. Analyze and compare their tangential velocities, angular velocities, and normal accelerations based purely on kinematic principles and the inverse-square law of gravity.

A roller coaster car is designed to travel through a vertical loop. The safety of the passengers depends heavily on the kinematics of the car, particularly its velocity and acceleration at the top and bottom of the loop. Describe the critical kinematic conditions required for the car to successfully navigate the loop without the passengers experiencing zero normal force (feeling "weightless") at the top, and explain the role of tangential and normal components of acceleration.