A uniform stick has a mass of \(4.42 \mathrm{~kg}\) and a length of \(1.23 \mathrm{~m}\). It is initially lying flat at rest on a frictionless horizontal surface and is struck perpendicularly by a puck imparting a horizontal impulsive force of impulse \(12.8 \mathrm{~N} \cdot \mathrm{s}\) at a distance of \(46.4 \mathrm{~cm}\) from the center. Determine the subsequent motion of the stick.

Angular velocity is a measure of how fast an object rotates or spins around a certain axis. It's commonly represented by the symbol \( \omega \) and calculated in radians per second (rad/s). When an object is struck off-center, as we saw in our exercise with the stick, it gains both linear and angular momentum. To calculate the angular velocity resulting from the impulse, you use the equation \( \omega = rI / I_0 \), where \( r \) is the distance to the point of impulse, \( I \) is the impulse, and \( I_0 \) is the moment of inertia. In the same manner as the linear velocity is affected by mass (inertia in linear motion), angular velocity is impacted by the distribution of an object's mass around the rotation axis, encapsulated in the moment of inertia term.

Understanding angular velocity is crucial in various real-life scenarios such as determining the speed of wheels turning on a vehicle, calculating rotational speeds in machinery, and even in analyzing the orbits of celestial bodies.

Linear velocity refers to the rate at which an object moves along a path in a particular direction. In physics, it's often denoted by \( v \) and measured in meters per second (m/s). Linear velocity can be calculated using the impulse-momentum theorem, which states that the impulse applied on an object is equal to the change in momentum of that object. The impulse, denoted as \( I \), is the product of the force applied and the time period over which it acts. Hence, for the stick in our exercise, finding the linear velocity after an off-center hit involves the simple formula of impulse divided by mass, \( v = I / m \).

This concept is widely applied in fields ranging from traffic accident analysis to sports physics, and is essential for understanding how objects move in space in response to forces.

The moment of inertia, symbolized by \( I_0 \), is a physical quantity expressing an object's tendency to resist angular acceleration. It is analogous to mass in linear motion and depends on both the mass of the object and the distribution of this mass relative to the axis of rotation. For regular shapes, the moment of inertia can be calculated using standard equations. For instance, in our exercise with the stick which can be considered a slender rod, the moment of inertia about an axis through its center is given by \( I_0 = (1/12)mL^2 \) where \( m \) is the mass and \( L \) is the length of the rod.

Understanding the moment of inertia is important for analyzing the dynamics of rotational motion in mechanical engineering, structural designs, and even in athletics, for equipment designs like golf clubs or discuses.

The impulse-momentum theorem is a fundamental principle in physics that describes the relationship between impulse and the momentum of an object. The theorem states that the impulse imparted to an object equals the change in momentum of the object. The equation representing this theorem can be written as \( I = \Delta p \), where \( I \) is the impulse, and \( \Delta p \) is the change in momentum. Momentum, in turn, equals mass times velocity, \( p = mv \). The impulse is a vector quantity, as is momentum, so direction matters and indicates that the force applied will cause motion in the direction of the applied force.

For our stick hit by the puck, the impulse leads to both linear motion and rotation. The theorem aids in understanding various phenomena such as the recoiling of a gun when fired, the landing of an athlete from a jump, and the effects of collisions in automotive safety.