Angular Momentum of a Particle Moving in a Straight Line

Angular momentum about a point, L = r x p, only cares about one thing beyond mass and speed: how far off-axis the motion is. Write it as L = m v r_perp, where r_perp is the perpendicular distance from your chosen reference point to the line the object is traveling along, extended infinitely in both directions. That distance never changes as the particle moves, even though r itself (the vector from the point to the particle) is constantly changing length and direction. That is why a particle sailing past you in a perfectly straight line has constant angular momentum about you the whole time, from far away, to closest approach, to far away again.

The mechanism is the cross product itself. r x p = r p sin(theta), and r sin(theta) is exactly the perpendicular distance from the reference point to the line of motion. As the particle approaches, r shrinks but theta grows toward 90 degrees, and the product r sin(theta) stays fixed at r_perp. Only at the instant of closest approach does r itself equal r_perp, with velocity exactly perpendicular to r.

This idea is what makes angular momentum conservation work for orbits and scattering problems later in the unit. A planet or comet moving under gravity would travel in a straight line if the force vanished, and it carries angular momentum about the sun throughout, which is exactly why that angular momentum stays constant as gravity bends the path into a curve. Seeing straight-line angular momentum first makes the conservation law in orbital motion feel inevitable rather than mysterious.