Does a particle moving at constant speed in a straight line have angular momentum about a point on the line? About a point not on the line? In either case, is its angular momentum constant?

In the realm of physics, the concept of linear momentum is fundamental. It is defined as the product of a particle's mass and its velocity. Mathematically, it is expressed as \( p = mv \), where \( m \) stands for mass and \( v \) for velocity. Linear momentum is a vector quantity, meaning it has both magnitude and direction. It reflects how much motion the particle has and in which direction it's moving.

When a particle moves at a constant speed in a straight line, its linear momentum remains constant both in magnitude and direction. However, if either the speed or direction changes, its momentum will change accordingly. This ties in with the law of conservation of momentum, which states that in a closed system, the total linear momentum remains constant unless acted upon by an external force. This principle plays a crucial role when analyzing scenarios involving collisions and interactions between bodies.

The cross product, also known as the vector product, is an operation on two vectors in three-dimensional space. It is denoted and computed as \( \textbf{a} \times \textbf{b} \), where \( \textbf{a} \) and \( \textbf{b} \) are vectors. The result of a cross product is a vector that is perpendicular to both \( \textbf{a} \) and \( \textbf{b} \), and its magnitude is equal to the area of the parallelogram that the vectors span when placed tail-to-tail.

The cross product is significant in calculating angular momentum, torque, and in determining the orientation of the resulting vector. For a particle moving in a straight line, the cross product of its position vector \( r \) with its linear momentum \( p \) gives the angular momentum \( L \) with respect to a point. If \( r \) and \( p \) are parallel or anti-parallel, the result is zero, indicating that the particle has no angular momentum about that point.

A particle is said to move with a constant speed if it covers equal distances in equal intervals of time, regardless of the direction of its motion. This concept is integral to understanding uniform motion in physics. When we say that a particle maintains a constant speed, we're indicating that its speed (the magnitude of its velocity) does not change over time.

In the given exercise, the particle’s motion at a constant speed plays a pivotal role in determining its angular momentum about various points. At constant speed, the linear momentum of the particle also remains constant since the particle’s mass does not change. This constancy allows us to conclude that if there's no external force acting to change the particle’s speed or direction, its angular momentum with respect to any reference point not on its line of motion remains constant. This applies only as long as the distance from the reference point to the particle and the angle between the velocity and the reference line do not change.